Non-zero and dynamic Stackelberg pursuit game control method for spacecraft system

Abstract

The invention provides a non-zero and dynamic Stackelberg pursuit game control method for a spacecraft system, and aims to solve the pursuit game problem of a spacecraft in the fields of space exploration, military reconnaissance and the like. According to the method, a traditional zero-sum game model is broken through, a non-zero-sum dynamic game framework is adopted, and the cooperation and competition relation between purchasers and escapers is reflected more practically. Effective control of a spacecraft pursuit game is realized through five steps of establishing a mathematical model, designing a control strategy, converting a cost function, solving a matrix Riccati equation and solving an optimal control gain.

Description

Technical Field

[0001] The present invention relates to the field of spacecraft control systems, and in particular to a non-zero-sum dynamic Stackelberg pursuit-escape game control method for a spacecraft system. Background Art

[0002] With the rapid development of aerospace technology, spacecraft are playing an increasingly important role in many fields such as space exploration, satellite communications, and military reconnaissance. At the same time, the pursuit and escape game between spacecraft has gradually become prominent. In mission scenarios such as satellite formation flying, space debris cleaning, and space confrontation, how to achieve efficient pursuit and successful escape has become a key technical challenge.

[0003] When modeling the spacecraft pursuit game problem, traditional control methods usually construct it as a zero-sum game in which the sum of the cost function of the pursuer and the cost function of the escaper is zero, that is, the gain of one party must be equal to the loss of the other party. However, in actual situations, the goals of the pursuer and the escaper are not completely opposite, which leads to overly radical strategy design, ignores safety and the reasonable allocation of resources, and limits the flexibility of strategy design. In some cases, the zero-sum framework will lead to moral and ethical dilemmas because it encourages the absolute victory of one party without considering the loss of the other party. In actual spacecraft pursuit, information is often incomplete, and the zero-sum framework cannot provide effective strategies under incomplete information. Compared with the zero-sum framework, the non-zero-sum framework can design more flexible and diverse strategies. The cost functions of the pursuer and the escaper do not have to be completely opposite, which is more in line with the actual situation. It can more accurately describe the complex relationship of cooperation and competition between the two parties in the game process to adapt to different mission requirements and environmental changes. It can take into account the possibility of long-term cooperation and win-win, which is more important for establishing a long-term and stable spacecraft pursuit strategy, so as to achieve a more stable and mutually beneficial strategy.

[0004] Another key problem with the current spacecraft pursuit game is that most existing solutions are based on static Nash games in game theory. The application of the traditional Nash equilibrium model in the spacecraft pursuit game has limitations because it fails to fully reflect the dynamic relationship of the order of decision-making in practice. In response to the above problems, under the Stackelberg dynamic game framework proposed by the present invention, the pursuer and the evader in the spacecraft pursuit game are no longer in a completely symmetrical position, but are based on the actual order and strategic advantages. This dynamic game process is more in line with the actual situation and can more accurately describe the complex relationship in the spacecraft pursuit game. Summary of the invention

[0005] The present invention provides a non-zero-sum dynamic Stackelberg pursuit-escape game control method for a spacecraft system, which regards the pursuer and evader of a spacecraft as the follower and leader in the Stackelberg pursuit-escape game, aiming to solve the inadequacy of the processing of the spacecraft pursuit-escape game problem in the prior art, especially in practical application scenarios with dynamic decision-making sequences. The present invention provides a new control strategy by constructing a non-zero-sum dynamic Stackelberg game model to achieve more effective decision-making of pursuit and evasion behaviors.

[0006] The present invention discloses a non-zero-sum dynamic Stackelberg pursuit-escape game control method for a spacecraft system, and the specific steps are as follows:

[0007] Step 1) Establish a mathematical model: Establish a mathematical model of the spacecraft pursuit and escape game. The system includes a pursuing spacecraft and an escaping spacecraft. Based on the motion state of the spacecraft, a pursuit and escape strategy model of the spacecraft system is constructed. In order to evaluate the effectiveness of the pursuit and escape strategy, a corresponding cost function is defined.

[0008] Step 2) Design control strategy: Combine the spacecraft pursuit relative motion equation and Stackelberg game theory to design a dynamic Stackelberg pursuit game control strategy.

[0009] Step 3) Transform the cost function: In order to simplify the solution of the problem, appropriate mathematical transformation is performed on the cost function to transform the cost function into an unconstrained optimization problem, making the problem easier to solve by numerical methods.

[0010] Step 4) Solve the matrix Riccati equation: The matrix Riccati equation corresponding to the pursuit spacecraft is obtained through the cost function in step 3), and the matrix Riccati equation is solved according to the boundary conditions.

[0011] Step 5) Solve the optimal control gain: Verify the reversibility of the optimal control gain matrix to ensure the effectiveness of the control strategy, and find the optimal control gain through an iterative calculation method.

[0012] The present invention proposes a non-zero-sum dynamic Stackelberg pursuit-escape game control method applicable to a spacecraft system and its specific steps. In view of the asynchronous characteristics of information acquisition and decision-making between the pursuer and the evader in the spacecraft system, the spacecraft pursuit-escape game framework is optimized and improved by means of the Stackelberg game in game theory, and the solution of the game equilibrium is solved by solving the matrix Riccati equation, and the optimal control decision of the pursuit-escape game is further obtained, thereby effectively improving the efficiency and adaptability of the spacecraft pursuit-escape game in practical application scenarios.

[0013] Compared with the prior art, the present invention has the following advantages:

[0014] Firstly, a new modeling method for spacecraft pursuit and escape game is proposed, which expands the traditional zero-sum game to a non-zero-sum game, which is closer to the interest relationship between the pursuer and the escapee in practical applications. Secondly, under the condition of complete information, a dynamic Stackelberg pursuit and escape game control strategy is designed, which allows the optimal decision to be made when the other party's decision is known, thus improving the scientificity and effectiveness of the decision. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 It is a control flow chart of the pursuit-escape game based on the non-zero-sum dynamic Stackelberg game of the present invention;

[0016] Figure 2 It is a simulation effect diagram of the spacecraft pursuit and escape game in the present invention;

[0017] FIG3 is a simulation effect diagram of a comparative experiment using other strategy methods. DETAILED DESCRIPTION

[0018] In order to better understand the technical solution of the present invention, the technical solution in the embodiment of the present invention will be described in detail below with reference to the accompanying drawings. Figure 1 As shown, the specific process of the present invention is as follows:

[0019] Step 1) Establish a mathematical model: Establish a mathematical model of the spacecraft pursuit game. The relative motion equation of the spacecraft in the Euler-Hill reference system is:

[0020] where x, y, z are the positions of the spacecraft in the Euler-Hill coordinate system; a x ,a y ,a z Along O x ,O y ,O z Direction control input, Where μ is the Earth's gravitational constant and a0 is the reference orbit radius.

[0021] The relative dynamic equation of the spacecraft pursuit-escape game is obtained by discretization: in Where T is the sampling period and T>0, and They are used to represent the status of the chasing spacecraft and the escaping spacecraft, and the status error a p,t and a e,t denote the control inputs of the chasing spacecraft and the escaping spacecraft respectively, t denotes the time step,

[0022] The cost function J of the pursuerp :

[0023] Cost function J for the escapee e : Where V N,P ,V P ,W 1,P ,W 1,e ,V N,e ,V e ,W 2,e ,W 2,p is a positive definite matrix with a certain dimension, is the state error at time t = N;

[0024] Step 2) Design control strategy: In the pursuit-escape game, the expected goal is to solve the optimal pursuit strategy and the optimal escape strategy According to the Stackelberg game theory, the spacecraft pursuit control strategy is designed as follows: “ST” is the abbreviation of “subject to”, which means operations or optimizations performed under certain restrictions or conditions.

[0025] Step 3) Convert the cost function: Convert the cost function into an unconstrained optimization problem. For ease of calculation, define where {P p,t ,P e,t} 0≤t≤N is a set of positive definite matrices, and the cost function of chasing the spacecraft is:

[0026] The cost function of escaping a spacecraft is:

[0027] Step 4) Solve the matrix Riccati equation: For the chasing spacecraft, the Riccati equation is derived as:

[0028] Its boundary conditions are: p,N =V N,p ; For the escaping spacecraft, the Riccati equation is derived as: The boundary conditions are: e,N =V N,e .

[0029] By solving the matrix Riccati equation, the optimal control gain matrix for the orbital maneuvers of the pursuit spacecraft and the escape spacecraft is obtained.

[0030] Step 5) Solve the optimal control gain: First construct the optimal control gain matrix Ψ t :

[0031] If t The optimal control gain of the Stackelberg pursuit game is obtained if it is reversible. Let t be recursively extended from t = 0 to t = N-1, and output the optimal control gain

[0032] Through the above steps, the chasing spacecraft can achieve the pursuit of the escaping spacecraft. Figure 2 In the figure, the state error is described. The state error is the state difference between the escaping spacecraft and the pursuing spacecraft as the number of iterations changes, and includes the relative position and relative speed information of the pursuing spacecraft. Figure 3 is a simulation effect diagram of the present invention, including a comparative experiment with the zero-sum game Nash equilibrium strategy and the non-zero-sum Nash equilibrium strategy under the same conditions: Figure 3a , compare and track the changing pattern of spacecraft acceleration. Figure 3b , compare and track the time variation of the spacecraft cost function. Figure 3c , comparing and tracking the spacecraft fuel consumption curve over time. Figure 3d , and compare how the relative distance between the two spacecraft changes over time. Figure 3e , compare the changing patterns of the relative speeds of the two spacecraft.

Claims

1. A non-zero-sum dynamic Stackelberg pursuit-escape game control method for a spacecraft system, characterized in that: The specific steps include: ⑴ Establish a mathematical model: Establish a mathematical model of the spacecraft pursuit and escape game. The system includes the pursuit spacecraft and the escape spacecraft. Based on the motion state of the spacecraft, a pursuit and escape strategy model of the spacecraft system is constructed. In order to evaluate the effectiveness of the pursuit and escape strategy, the corresponding cost function is defined; ⑵ Design control strategy: Combine the spacecraft pursuit and escape relative motion equation and Stackelberg game theory to design dynamic Stackelberg pursuit and escape game control strategy; ⑶ Transform the cost function: In order to simplify the solution of the problem, the cost function is appropriately transformed mathematically and transformed into an unconstrained optimization problem, making it easier to solve the problem by numerical methods; (4) Solve the matrix Riccati equation: The matrix Riccati equation corresponding to the pursuit spacecraft is obtained through the cost function, and the matrix Riccati equation is solved according to the boundary conditions; ⑸ Solve the optimal control gain: Verify the reversibility of the optimal control gain matrix to ensure the effectiveness of the control strategy, and obtain the optimal control gain through an iterative calculation method.

2. The method according to claim 1, characterized in that: The step (1) comprises: the relative motion equation of the spacecraft in the Euler-Hill reference frame: where x, y, z are the positions of the spacecraft in the Euler-Hill coordinate system, and a x ,a y ,a z Along O x ,O y ,O z Direction control input, Where μ is the earth's gravitational constant, a0 is the reference orbit radius, and the relative dynamic equation of the spacecraft pursuit-escape game is obtained by discretization: Where X d =e XT , Where T is the sampling period and T>0, and They are used to represent the status of the chasing spacecraft and the escaping spacecraft, and the status error a p,t and a e,t denote the control inputs of the chasing spacecraft and the escaping spacecraft respectively, t denotes the time step, The cost function J of the pursuer p : Cost function J for the escapee e : Where V N,P ,V P ,W 1,P ,W 1,e ,V N,e ,V e ,W 2,e ,W 2,p is a positive definite matrix with a certain dimension, is the state error at time t=N.

3. The method according to claim 1, characterized in that: The step (2) includes: in the pursuit-escape game, the expected goal is to solve the optimal pursuit strategy and the optimal escape strategy According to the Stackelberg game theory, the spacecraft pursuit control strategy is designed as follows: "st" is the abbreviation of "subject to", which means operations or optimizations performed under certain restrictions or conditions.

4. The method according to claim 1, characterized in that: The step (3) includes: transforming the cost function into an unconstrained optimization problem. For the convenience of calculation, define where {P p,t ,P e,t } 0≤t≤N is a set of positive definite matrices, and the cost function of chasing the spacecraft is: The cost function of escaping a spacecraft is:

5. The method according to claim 1, characterized in that: The step (4) comprises: for the chasing spacecraft, deriving the Riccati equation as: Its boundary conditions are: p,N =V N,p , for the escaping spacecraft, the Riccati equation is derived as: The boundary conditions are: e,N =V N,e ,By solving the matrix Riccati equation, the optimal control gains for the orbital maneuvers of the pursuit spacecraft and the escape spacecraft are obtained as follows.

6. The method according to claim 1, characterized in that: The step (5) includes: to solve the optimal control gain of the pursuit-escape game, first construct the optimal control gain matrix Ψ t : If t Reversible can get the optimal control gain of the Stackelberg pursuit game Let t be recursively extended from t = 0 to t = N-1, and output the optimal control gain

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