Differential game interception control method for spacecraft under incomplete information
By establishing relative dynamic equations and game index functions during spacecraft interception, estimating the target star control matrix with behavioral learning algorithms, designing the thrust game strategy of intercepting stars, the problem of spacecraft interception strategy deviating from the actual situation under incomplete information is solved, and a fast and adaptive interception effect is achieved.
Patent Information
- Application Number
- CN202311066870.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-08-23
- Publication Date
- 2025-08-29
- Estimated Expiration
- 2043-08-23
AI Technical Summary
In the existing technology, under incomplete information conditions, spacecraft interception strategies cannot effectively adapt to target maneuvers, resulting in the inability to achieve rapid interception.
The spacecraft differential game interception control method under incomplete information is adopted. By establishing the relative dynamic equations between the interceptor star and the target star, the game index function is determined, the symmetric positive definite matrix P is set, and the target star control matrix is estimated in combination with the behavioral learning algorithm, and the thrust game strategy for the interceptor star is designed.
It realizes rapid interception of targets under incomplete information conditions, the control strategy is close to the actual situation, and has adaptive and high-precision target information estimation.
Smart Images
Figure CN117022676B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to a spacecraft differential game interception control method under incomplete information, and belongs to the field of spacecraft game control. Background Art
[0002] In the satellite interception problem, when the interceptor satellite approaches the target satellite, the target satellite will adopt an evasive maneuver strategy. In this case, traditional unilateral optimization interception strategies are no longer applicable, and it is necessary to study bilateral strategies that consider target maneuvers. However, due to battlefield environmental constraints or sensor limitations, the interceptor may not be able to fully obtain target information, resulting in an interception situation with incomplete information. Therefore, studying spacecraft game interception control methods under incomplete information is an important step in the development of space pursuit and escape problems.
[0003] Differential games (ISAACS R. Differential Games [M]. New York: John Wiley and Sons, 1965: 1-5.) involve a process of confrontation between two players in which at least one player can use state information from previous processes to determine its current action. If the players' goals are not completely aligned, the game is considered non-cooperative. Because game theory can simultaneously consider the control variables of multiple spacecraft, its application to the pursuit-and-escape problem has attracted considerable attention.
[0004] Hafer WT, Reed H L. Orbital pursuit-evasion hybrid spacecraft controllers [C] considered the bilateral game of spacecraft pursuit and evasion, and the strategies of each task. The game value function and the trajectory boundary were used to determine whether to switch strategies, thus achieving a balance between the pursuit and evasion tasks and the respective tasks. Potani M, Conway BA. Optimal Interception of Evasive Missile Warhead Numerical Solution of the Differential Game [J]. Journal of Guidance, Control and Dynamics, 2008, 31 (4): 1111-1122. The spacecraft long-range interception game was studied. In order to solve the problem of the difficulty in determining the initial values of the co-state variables, the approximate initial values were solved by genetic algorithm optimization. Then the motion trajectory was discretized and the states of each discrete point were configured. The exact values of the initial values of the co-state variables were solved by nonlinear programming optimization.
[0005] Due to the non-cooperative nature of the target, its information is usually not fully available. To address this interception problem, Prokopov O and Shima T. Linear quadratic optimal cooperative strategies for active aircraft protection [C] studied the target escape defense problem. Given an interceptor strategy, the optimal game strategy between the target and the defender was established under the scenarios of one-way communication and two-way communication between the target and the defender. Cavalieri KA. Incomplete information pursuit-evasion games with application to spacecraft rendezvous and missile defense [D] used double integral dynamics as a dynamic model to study the cases of incomplete target information and imperfect dynamics information. Both incomplete and imperfect information were considered as extended state variables, and the original dynamics were augmented. By designing an observer, the extended state was estimated, and thus the incomplete and imperfect information were estimated.
[0006] While there has been extensive research on the game-based interception problem, existing models for spacecraft pursuit and escape are still incomplete. Most models assume that the interceptor has complete information about the target, or use incomplete information games based on simplified dynamics. The resulting control strategies may deviate from the actual situation and fail to meet the requirements for rapid interception. Summary of the Invention
[0007] Aiming at the problem that incomplete target information causes the game interception strategy to deviate from the actual situation and fail to achieve rapid interception during spacecraft pursuit, the present invention provides a spacecraft differential game interception control method under incomplete information.
[0008] The present invention provides a spacecraft differential game interception control method under incomplete information, comprising:
[0009] Step 1: Establish a reference satellite. In the orbital coordinate system with the reference satellite as the origin, establish the dynamic equation of the interceptor satellite relative to the reference satellite and simplify it into a CW equation. Then, convert the CW equation into a state-space equation and expand it to obtain the expansion of the relative dynamic equation between the interceptor satellite and the target satellite.
[0010] Step 2: Determine the game index function of the interceptor satellite and the target satellite under the optimal control strategy based on the expansion of the relative dynamic equation of the interceptor satellite and the target satellite;
[0011] Step 3: Set a symmetric positive definite matrix P, and combine the optimal control strategy of the game indicator function to establish a saddle point strategy pair that satisfies the Nash equilibrium between the interceptor and the target star; and determine the constraints of the symmetric positive definite matrix P;
[0012] Step 4: Use the Epsilon Nash equilibrium pair to describe the saddle point strategy pair, and then use the behavioral learning algorithm to estimate the target star control matrix;
[0013] Step 5: Based on the estimated value of the target satellite control matrix, the symmetric positive definite matrix P is determined by the constraints of the symmetric positive definite matrix P, and then the thrust game strategy of the interceptor satellite is obtained based on the symmetric positive definite matrix P. The thrust game strategy is used to intercept and control the target satellite.
[0014] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 1, the reference satellite is taken as the origin O1, the direction of the geocentric radial vector is the x-axis, the direction of the orbital angular momentum is the z-axis, and the y-axis satisfies the right-hand rule, and the orbital coordinate system O1xyz is defined.
[0015] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 1, the CW equation is:
[0016]
[0017]
[0018]
[0019] Where x1 is the x-axis coordinate of the interceptor satellite relative to the reference satellite, y1 is the y-axis coordinate of the interceptor satellite relative to the reference satellite, z1 is the z-axis coordinate of the interceptor satellite relative to the reference satellite; ω is the orbital angular velocity of the reference satellite; is the x-axis thrust of the interceptor, is the y-axis thrust of the interceptor, is the z-axis thrust of the interceptor star.
[0020] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 1, the CW equation is converted into a state space equation:
[0021]
[0022] Where X is the interceptor satellite state variable: U is the control thrust:
[0023] A and B are intermediate variables:
[0024]
[0025] Expand the state space equation to obtain the relative dynamic equation expansion between the interceptor and target satellites;
[0026]
[0027] Where X PE is the relative state between the interceptor star and the target star, U P The thrust game strategy for intercepting satellites, U E is the thrust interception strategy of the target satellite, C is the intermediate variable, C=B.
[0028] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 2, the optimal control strategy is:
[0029] In the game process, the interceptor satellite tends to intercept the target satellite quickly at the minimum cost, and the target satellite tends to increase the distance between it and the interceptor satellite as much as possible at the minimum cost;
[0030] Define the game indicator function J as:
[0031]
[0032] Where t f is the terminal moment, S is the intermediate variable, and is a symmetric positive definite matrix; t0 is the starting moment, Q is a symmetric semi-positive definite matrix, and R P is the interceptor satellite control matrix, R E is the target star control matrix, and t is the time.
[0033] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 3, the saddle point strategy pair of the interceptor satellite and the target satellite that satisfies the Nash equilibrium is:
[0034]
[0035] The symmetric positive definite matrix P satisfies the final value condition P(t f )=S,
[0036] The thrust amplitude is limited to: ||U P ||≤ρ P ,||U E ||≤ρ E ,
[0037] where ρ P is the maximum thrust amplitude of the interceptor satellite, ρ E is the maximum thrust amplitude of the target satellite;
[0038] At the same time, the symmetric positive definite matrix P satisfies the Riccati differential equation:
[0039]
[0040] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 4, the method for obtaining the estimated value of the target satellite control matrix is:
[0041] Assume that the target star control matrix R E remains unchanged in the current cycle's interception;
[0042] Define extended state variable Y = [X PE r E ] T , where r E is the target star control matrix information value,
[0043] get:
[0044]
[0045] Where f(Y) is the intermediate function, Z is the measured state variable, and I6 is the 6th-order unit matrix;
[0046] Discretize the above formula to get the extended state variable deviation value ΔY at time k k and the measured state variable deviation ΔZ at time k k :
[0047]
[0048] Where Φ(k,k-1) is the state transfer matrix at time k, and Φ(k,k-1)≈I+F n T, where F n is the Jacobian matrix, T is the sampling time;
[0049] W k-1 is the k-1 moment process noise, H k is the measurement matrix at time k, V k is the measurement noise at time k;
[0050] The following conditions must be met at the same time:
[0051]
[0052] Where W k is the process noise at time k, the variable with subscript j represents the corresponding variable at time j, Q k is the variance matrix of the system noise sequence at time k, which is a semi-positive definite matrix; R k is the variance matrix of the measurement noise sequence at time k, δ kj is the Kronecker symbol;
[0053] The generalized Kalman filter is used for state estimation to obtain the predicted value of the extended state variable deviation at time k And the predicted value P of the extended state variable error covariance matrix at time k k,k-1 :
[0054]
[0055] In the formula K is the estimated value of the expanded state variable deviation at time k, k is the filter gain matrix at time k,
[0056] P k is the estimated value of the extended state variable error covariance matrix at time k, and I is the identity matrix;
[0057] Based on and P k,k-1 Determine the relative state estimate of the interceptor satellite and the target satellite, and calculate the target satellite control matrix information value r E Estimated value of Then we can get the estimated value of the target star control matrix Then calculate the thrust game strategy U of the interceptor p .
[0058] According to the spacecraft differential game interception control method under incomplete information of the present invention, in step 5, the target satellite control matrix estimate is As the target star control matrix R E , the Riccati differential equation is used to calculate the symmetric positive definite matrix P:
[0059]
[0060] Then, based on the symmetric positive definite matrix P, the thrust game strategy of the interceptor satellite is calculated:
[0061]
[0062] Beneficial effects of the present invention: The method of the present invention studies the spacecraft game problem under incomplete information, obtains a differential game control strategy control method that satisfies the Epsilon Nash equilibrium, and realizes the interception of the target under incomplete information.
[0063] The control strategy application of the method of the present invention is closer to the actual situation; a behavior learning information estimation algorithm is used to estimate the target control strategy, with high valuation accuracy; adaptive interception can be guaranteed to meet the demand for rapid interception. BRIEF DESCRIPTION OF THE DRAWINGS
[0064] Figure 1 is a flow chart of the spacecraft differential game interception control method under incomplete information according to the present invention;
[0065] Figure 2is a schematic diagram of the orbital coordinate system; in the figure, OXYZ is the inertial coordinate system, r1 is the position vector radius of the reference satellite and the center of the earth, r2 is the position vector radius of the interceptor satellite P0 and the center of the earth, and δr is the relative position vector radius of the interceptor satellite and the reference satellite;
[0066] Figure 3 is the thrust game strategy U of the interceptor star P Flowchart of the calculation process;
[0067] In the picture is the estimated value of the relative state variables between the interceptor and target satellites at time k-1, P k-1 / k-1 is the estimated value of the extended state variable error covariance matrix at time k-1, which is equivalent to P k-1 ;z k is the observed value of the measurement state at time k, is the estimated value of the relative state variables between the interceptor and target satellite at time k, P k / k is the estimated value of the extended state variable error covariance matrix at time k, which is equivalent to P k ; is the estimated value of the target star control matrix information at time k;
[0068] Figure 4 is a pursuit trajectory diagram of the intercepting spacecraft (interceptor satellite) and the target spacecraft (target satellite) in the orbital coordinate system in a specific embodiment;
[0069] Figure 5 This is a diagram showing the change in relative distance between the interceptor satellite and the target satellite when the control method of the present invention is used in a specific embodiment;
[0070] Figure 6 It is a three-axis variation diagram of the control acceleration of the interceptor (interceptor satellite) when the method of the present invention is used for control in a specific embodiment. DETAILED DESCRIPTION
[0071] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.
[0072] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.
[0073] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.
[0074] Specific implementation method 1. Combination Figure 1 and Figure 2 As shown, the present invention provides a spacecraft differential game interception control method under incomplete information, comprising:
[0075] Step 1: Establish a reference satellite. In the orbital coordinate system with the reference satellite as the origin, establish the dynamic equation of the interceptor satellite relative to the reference satellite and simplify it into a CW equation. Then, convert the CW equation into a state-space equation and expand it to obtain the expansion of the relative dynamic equation between the interceptor satellite and the target satellite.
[0076] Step 2: Determine the game index function of the interceptor satellite and the target satellite under the optimal control strategy based on the expansion of the relative dynamic equation of the interceptor satellite and the target satellite;
[0077] Step 3: Set a symmetric positive definite matrix P, and combine the optimal control strategy of the game indicator function to establish a saddle point strategy pair that satisfies the Nash equilibrium between the interceptor and the target star; and determine the constraints of the symmetric positive definite matrix P;
[0078] Step 4: Use the Epsilon Nash equilibrium pair to describe the saddle point strategy pair, and then use the behavioral learning algorithm to estimate the target star control matrix;
[0079] Step 5: Based on the estimated value of the target satellite control matrix, the symmetric positive definite matrix P is determined by the constraints of the symmetric positive definite matrix P, and then the thrust game strategy of the interceptor satellite is obtained based on the symmetric positive definite matrix P. The thrust game strategy is used to intercept and control the target satellite.
[0080] During the terminal interception of a spacecraft, the interceptor satellite is relatively close to the target satellite. Therefore, a reference satellite is established near the interceptor satellite to establish the spacecraft dynamic equations while ignoring all perturbations. This direction involves two players: the target spacecraft and the interceptor spacecraft.
[0081] Further, combined Figure 2 As shown in Figure 1, in step 1, assume that the reference satellite is operating in a circular orbit, with the reference satellite as the origin O1, the direction of the Earth's radial vector as the x-axis, the direction of the orbital angular momentum as the z-axis, and the y-axis satisfying the right-hand rule, defining the orbital coordinate system O1xyz. In the orbital coordinate system O1xyz, the dynamic equation of the interceptor satellite relative to the reference satellite can be simplified to the CW equation:
[0082]
[0083]
[0084]
[0085] Where x1 is the x-axis coordinate of the interceptor satellite relative to the reference satellite, y1 is the y-axis coordinate of the interceptor satellite relative to the reference satellite, z1 is the z-axis coordinate of the interceptor satellite relative to the reference satellite; ω is the orbital angular velocity of the reference satellite; is the x-axis thrust of the interceptor, is the y-axis thrust of the interceptor, is the z-axis thrust of the interceptor star.
[0086] In step 1, the CW equation is converted into a state space equation:
[0087]
[0088] Where X is the interceptor satellite state variable: U is the control thrust:
[0089] A and B are intermediate variables:
[0090]
[0091] Expand the state space equation to obtain the relative dynamic equation expansion between the interceptor and target satellites;
[0092]
[0093] Where X PE is the relative state between the interceptor star and the target star, U P The thrust game strategy for intercepting satellites, U E is the thrust interception strategy of the target satellite, C is the intermediate variable, C=B.
[0094] X PE =X P -X E , where X P Equal to the intercept star state variables X, X E is the target star state variable. For X PE The first derivative of .
[0095] Going further, in step 2, the optimal control strategy is:
[0096] Determine the target function of each player. The target function is the goal that each spacecraft hopes to achieve.
[0097] In the game process, the interceptor satellite tends to intercept the target satellite quickly at the minimum cost, and the target satellite tends to increase the distance between it and the interceptor satellite as much as possible at the minimum cost;
[0098] Define the game indicator function J as:
[0099]
[0100] Where t fis the terminal moment, S is the intermediate variable, and is a symmetric positive definite matrix; t0 is the starting moment, Q is a symmetric semi-positive definite matrix, and R P is the interceptor satellite control matrix, R E is the target star control matrix, and t is the time.
[0101] Furthermore, in step 3, the saddle point strategy pair of the intercepting star and the target star that satisfies the Nash equilibrium is:
[0102]
[0103] The symmetric positive definite matrix P satisfies the final value condition P(t f )=S,
[0104] The thrust amplitude is limited to: ||U P ||≤ρ P ,||U E ||≤ρ E ,
[0105] where ρ P is the maximum thrust amplitude of the interceptor satellite, ρ E is the maximum thrust amplitude of the target satellite;
[0106] At the same time, the symmetric positive definite matrix P satisfies the Riccati differential equation:
[0107]
[0108] Combine Figure 3 As shown in Figure 2, in step 4, the method for obtaining the estimated value of the target star control matrix is:
[0109] In this embodiment, it is considered that the interceptor satellite cannot obtain the target satellite control matrix R E The situation is described by Epsilon Nash equilibrium, and the target information is estimated through the behavioral learning algorithm. The specific steps are as follows:
[0110] Intercept Star uses an estimator to estimate R E Estimation is performed, and the target star can obtain complete information. The control strategy is determined by U E This embodiment assumes that the target star control matrix R E remains unchanged in the intercept of the current sampling period;
[0111] Define extended state variable Y = [X PE r E ] T , where r E is the target star control matrix information value,
[0112] Substituting the target control strategy into the relative state equation, we get:
[0113]
[0114] In the formula is the first-order derivative of Y, f(Y) is the intermediate function, Z is the measured state variable, and I6 is the 6th-order unit matrix;
[0115] Discretize the above formula to get the extended state variable deviation value ΔY at time k k and the measured state variable deviation ΔZ at time k k :
[0116]
[0117] Where Φ(k,k-1) is the state transfer matrix at time k, and Φ(k,k-1)≈I+F n T, where F n is the Jacobian matrix, T is the sampling time;
[0118] W k-1 is the k-1 moment process noise, H k is the measurement matrix at time k, V k is the measurement noise at time k;
[0119] The following conditions must be met at the same time:
[0120]
[0121] Where W k is the process noise at time k, the variable with subscript j represents the corresponding variable at time j, Q k is the variance matrix of the system noise sequence at time k, which is a semi-positive definite matrix; R k is the variance matrix of the measurement noise sequence at time k, δ kj is the Kronecker symbol;
[0122] Determine the initial time step bias estimate of the expanded state variable and the estimated value of the extended state variable error covariance matrix P at the initial moment k-1 , use generalized Kalman filter (similar to EKF) for state estimation to obtain the predicted value of the extended state variable deviation at time k And the predicted value P of the extended state variable error covariance matrix at time k k,k-1 :
[0123] The filtering equation is:
[0124]
[0125] In the formula K is the estimated value of the expanded state variable deviation at time k, kis the filter gain matrix at time k,
[0126] P k is the estimated value of the extended state variable error covariance matrix at time k, and I is the identity matrix;
[0127] Based on and P k,k-1 Determine the relative state estimate of the interceptor satellite and the target satellite, and calculate the target satellite control matrix information value r E Estimated value of Then we can get the estimated value of the target star control matrix Then calculate the thrust game strategy U of the interceptor p .
[0128] Finally, in step five, after the estimation of the target state information is completed, the corresponding interception strategy is designed using the differential game control method to achieve adaptive interception of the target.
[0129] The target star control matrix estimate As the target star control matrix R E , the Riccati differential equation is used to calculate the symmetric positive definite matrix P:
[0130]
[0131] Recombination Figure 3 As shown in Figure 2, after obtaining the target state estimation information, the corresponding control strategy is designed, that is, the thrust game strategy of the interceptor satellite is calculated based on the symmetric positive definite matrix P:
[0132] Specific embodiment:
[0134] In order to verify the effectiveness of the game model established by the method of the present invention and the solution strategy adopted, a simulation calculation of the pursuit-escape model under incomplete information was carried out. The initial conditions are set as follows: Assume that the interceptor spacecraft and the target are both operating near the low Earth orbit (LEO), and select a satellite close to them in the low Earth orbit as the reference satellite, whose orbital angular velocity ω = 0.001 rad·s -1 The initial positions of the interceptor and the target are [1.5 0.5 0] T km, [0 0 0] T km, and the initial velocities are [0 0 0] T km·s -1 , [-0.05 0 0.05] T km·s -1 The game terminal time is 2000s. Assume that the maximum thrust acceleration per unit mass of the interceptor and the target is 10m / s. 2The true value of the control parameter in the indicator function is set as follows:
[0135] R P =1×10 6 I, R E =1.5×10 6 I;
[0136] After obtaining the target state estimation information, the corresponding control strategy is designed. The parameters of each filter are set as follows: The process noise variance matrix is diag{[10 -6 10 -6 10 -6 0.25×10 -6 0.25×10 -6 0.25×10 -6 10 10 ]}; the measurement noise variance matrix is [10 -8 10 -8 10 -8 0.25×10 -8 0.25×10 -8 0.25×10 -8 ] T .
[0137] The simulation results of this embodiment are as follows Figure 4-6 As shown in Figure 2: The interceptor takes the approach of designing a corresponding control strategy after obtaining the target state estimation information, estimates the target control matrix, and then establishes a game strategy. The relative distance between the interceptor and the target changes as shown in Figure 2. Figure 5 As shown in the figure, effective estimation of target information can be achieved by designing corresponding control strategies after obtaining target state estimation information.
[0138] In summary, the method of the present invention constructs a terminal interception game model by analyzing the problem of pursuing maneuverable spacecraft under incomplete information, and uses differential game theory to analyze and solve a control method, which provides the intercepting spacecraft with an optimal control strategy and optimal interception trajectory that is in line with its own interests. The model application is practical, has the ability of rapid interception, and is highly adaptable.
[0139] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be used in conjunction with other described embodiments.
Claims
1. A spacecraft differential game interception control method under incomplete information, characterized by include, Step 1: Establish a reference satellite. In the orbital coordinate system with the reference satellite as the origin, establish the dynamic equation of the interceptor satellite relative to the reference satellite and simplify it into a CW equation. Then the CW equation is transformed into a state space equation and expanded to obtain the relative dynamic equation expansion between the interceptor satellite and the target satellite; Step 2: Determine the game index function of the interceptor satellite and the target satellite under the optimal control strategy based on the expansion of the relative dynamic equation of the interceptor satellite and the target satellite; Step 3: Set a symmetric positive definite matrix P, and combine the optimal control strategy of the game indicator function to establish a saddle point strategy pair that satisfies the Nash equilibrium between the interceptor and the target star; and determine the constraints of the symmetric positive definite matrix P; Step 4: Use the Epsilon Nash equilibrium pair to describe the saddle point strategy pair, and then use the behavioral learning algorithm to estimate the target star control matrix; Step 5: Based on the estimated value of the target satellite control matrix, the symmetric positive definite matrix P is determined by the constraints of the symmetric positive definite matrix P. Then, the thrust game strategy of the interceptor satellite is obtained based on the symmetric positive definite matrix P, and the thrust game strategy is used to intercept and control the target satellite. In step 1, the reference satellite is taken as the origin O1, the direction of the Earth's center radial vector is the x-axis, the direction of the orbital angular momentum is the z-axis, and the y-axis satisfies the right-hand rule, defining the orbital coordinate system O1xyz; In step 1, the CW equation is: Where x1 is the x-axis coordinate of the interceptor satellite relative to the reference satellite, y1 is the y-axis coordinate of the interceptor satellite relative to the reference satellite, z1 is the z-axis coordinate of the interceptor satellite relative to the reference satellite; ω is the orbital angular velocity of the reference satellite; is the x-axis thrust of the interceptor, is the y-axis thrust of the interceptor, is the z-axis thrust of the interceptor star.
2. The spacecraft differential game interception control method under incomplete information according to claim 1 is characterized in that: In step 1, the CW equation is converted into a state space equation: Where X is the interceptor satellite state variable: U is the control thrust: A and B are intermediate variables: Expand the state space equation to obtain the relative dynamic equation expansion between the interceptor and target satellites; Where X PE is the relative state between the interceptor star and the target star, U P The thrust game strategy for intercepting satellites, U E is the thrust interception strategy of the target satellite, C is the intermediate variable, C=B.
3. The spacecraft differential game interception control method under incomplete information according to claim 2 is characterized in that: In step 2, the optimal control strategy is: In the game process, the interceptor satellite tends to intercept the target satellite quickly at the minimum cost, and the target satellite tends to increase the distance between it and the interceptor satellite as much as possible at the minimum cost; Define the game indicator function J as: Where t f is the terminal moment, S is the intermediate variable, and is a symmetric positive definite matrix; t0 is the starting moment, Q is a symmetric semi-positive definite matrix, and R P is the interceptor control matrix, R E is the target star control matrix, and t is the time.
4. The spacecraft differential game interception control method under incomplete information according to claim 3 is characterized in that: In step 3, the saddle point strategy pair of the intercepting star and the target star that satisfies the Nash equilibrium is: The symmetric positive definite matrix P satisfies the final value condition P(t f )=S, The thrust amplitude is limited to: ||U P ||≤ρ P ,||U E ||≤ρ E , where ρ P is the maximum thrust amplitude of the interceptor satellite, ρ E is the maximum thrust amplitude of the target satellite; At the same time, the symmetric positive definite matrix P satisfies the Riccati differential equation:
5. The spacecraft differential game interception control method under incomplete information according to claim 4 is characterized in that: In step 4, the method for obtaining the estimated value of the target star control matrix is: Assume that the target star control matrix R E remains unchanged in the current cycle's interception; Define extended state variable Y = [X PE r E ] T , where r E is the target star control matrix information value, get: Where f(Y) is the intermediate function, Z is the measured state variable, and I6 is the 6th-order unit matrix; Discretize the above formula to get the extended state variable deviation value ΔY at time k k and the measured state variable deviation ΔZ at time k k : Where Φ(k,k-1) is the state transfer matrix at time k, and Φ(k,k-1)≈I+F n T, where F n is the Jacobian matrix, T is the sampling time; W k-1 is the k-1 moment process noise, H k is the measurement matrix at time k, V k is the measurement noise at time k; The following conditions must be met at the same time: Where W k is the process noise at time k, the variable with subscript j represents the corresponding variable at time j, Q k is the variance matrix of the system noise sequence at time k, which is a semi-positive definite matrix; R k is the variance matrix of the measurement noise sequence at time k, δ kj is the Kronecker symbol; The generalized Kalman filter is used for state estimation to obtain the predicted value of the extended state variable deviation at time k And the predicted value P of the extended state variable error covariance matrix at time k k,k-1 : In the formula K is the estimated value of the expanded state variable deviation at time k, k is the filter gain matrix at time k, P k is the estimated value of the extended state variable error covariance matrix at time k, and I is the identity matrix; Based on and P k,k-1 Determine the relative state estimate of the interceptor satellite and the target satellite, and calculate the target satellite control matrix information value r E Estimated value of Then we can get the estimated value of the target star control matrix Then calculate the thrust game strategy U of the interceptor p .
6. The spacecraft differential game interception control method under incomplete information according to claim 5 is characterized in that: In step 5, the target star control matrix estimate As the target star control matrix R E , the Riccati differential equation is used to calculate the symmetric positive definite matrix P: Then, based on the symmetric positive definite matrix P, the thrust game strategy of the interceptor satellite is calculated: