A Model-Based Predictive Method for Leaderless Formation Control Based on Distributed Evolutionary Game Theory

Abstract

This invention proposes a leaderless formation control method based on distributed evolutionary game theory, which overcomes the shortcomings of leader-follower formation control algorithms. This invention employs a leaderless formation control algorithm, where all agents have the same role and function. It utilizes model predictive control to construct a global optimization problem, designing a formation error function within the global model prediction cost function to achieve formation. Collision avoidance is implemented by constructing a safe distance set for each agent using a Voronoi diagram. This transforms the formation control problem into an evolutionary game problem, enabling distributed solution. Furthermore, the invariant set property of evolutionary games ensures that no collision occurs during movement. Moreover, this invention is also applicable to time-varying communication networks, improving control and safety performance while reducing computational complexity and communication burden.

Description

Technical Field

[0001] This invention belongs to the field of multi-agent formation control technology, specifically relating to a leaderless formation control method based on distributed evolutionary game theory model prediction. Background Technology

[0002] In recent years, with the continuous development of multi-agent systems, formation control has become a hot research topic in multi-agent systems. Formation control refers to the process by which multiple agents, such as unmanned vehicles and drones, maintain their desired positions while adapting to environmental constraints (such as obstacle avoidance) as they move towards a target location. It enables the completion of specific complex tasks without human intervention, and thus has been widely applied in various fields such as military, aerospace, and industry, showing promising development prospects. However, a major challenge in practical applications of multi-agent formation control is that all agents must possess the ability to avoid collisions with obstacles or other agents, and the communication topology may change over time during agent movement. Furthermore, when forming a formation in a distributed manner, each agent needs to know the state of other agents, but communication between agents may cease when the communication topology changes.

[0003] The leader-follower control method, as a solution to the current formation control problem, is based on the principle of assigning one agent as the leader to track a reference trajectory, while the other agents act as followers, maintaining a certain distance from the leader, thus achieving formation control. Due to its simplicity, it is widely used in multi-agent formations. However, the leader-follower formation problem has the following two drawbacks: 1) The entire system is overly reliant on the leader; when the leader fails to track the reference trajectory, the entire multi-agent formation will deviate from the reference trajectory; 2) The leader agent does not take into account the following behavior of the followers, which may lead to situations where the leader agent moves too fast, and the followers cannot keep up. Summary of the Invention

[0004] In view of this, the present invention provides a distributed model prediction leaderless formation control method based on distributed evolutionary game theory, in which all agents have the same roles and functions, and can form a formation without collisions by each agent only needing to obtain local information from its neighbors under communication constraints.

[0005] To achieve the above objectives, this invention provides a distributed model prediction method for leaderless group formation control based on distributed evolutionary game theory, comprising the following steps:

[0006] Step 1: Establish a multi-agent system, define the initial and target positions of the agents, construct a dynamic model of the agents, and solve the optimal control problem involving obstacle avoidance constraints, control constraints, and state constraints among the agents. The optimization problem is to predict the state of the agents in the future using a prediction model, given that the final target state is known, so as to minimize the distance between the agent's position and the target position in the future, and obtain the optimal control input at the current moment.

[0007] Step 2: Create a safe distance set for each agent to ensure that each agent will not collide as long as it moves within the specified safe distance set;

[0008] Step 3: Propose two group evolutionary games under coupling constraints, select a modification protocol to construct evolutionary dynamic equations, so that the evolutionary dynamic equations of each population can reach the Nash equilibrium solution of the game through continuous iteration and optimization, and have the property of invariant set.

[0009] Step 4: Transform the constructed multi-agent formation problem into two group evolutionary game problems with coupling constraints, and use the evolutionary dynamics equations of evolutionary games to solve the multi-agent formation optimization problem.

[0010] In step 4, the positions of agents in formation control are transformed into population states in evolutionary game, and each agent in formation control is transformed into a strategy in evolutionary game. The cost function in formation control problem and the benefit function in evolutionary game are combined, and then the optimal control problem in step 1 is solved using evolutionary dynamics equations.

[0011] The optimization problem in step 1 is as follows:

[0012] min u(k) J(k)

[0013] stform = 0, 1, ..., H p -1

[0014]

[0015]

[0016]

[0017]

[0018] in: This represents the position information of the i-th agent. This represents the speed information of the i-th agent. Let i represent the state variable of the i-th agent. This represents the control variable for the i-th agent. This represents the collision avoidance constraint set for the i-th agent. Indicates the movable range of multiple agents. This indicates the range of controllable outputs allowed for a single agent.

[0019] The set of safe distances in step 2 is defined as follows:

[0020]

[0021]

[0022]

[0023]

[0024] Where R is the specified safety distance, set It is a polyhedral closed set, for any and Satisfy‖c i (k)-c j (k)‖≥R, Let δ represent the set of neighboring agents of agent i. ij (k), ε ij (k) and ω ij (k) represents the intermediate variable used in the calculation.

[0025] In step 2, a distributed evolutionary game with two populations having coupling constraints is used. The specific steps are as follows: the optimization problem of the evolutionary game is solved by finding the Nash equilibrium point; the optimization problem of finding the Nash equilibrium point is substituted into the average dynamics to obtain the distributed Smith dynamics equations of the two populations with coupling constraints.

[0026] Beneficial effects:

[0027] 1. This invention extends the average dynamics of evolutionary game theory to the coupled constraints between two populations, and proves that this evolutionary dynamics, through continuous iteration and optimization, will eventually reach the Nash equilibrium point of the game. It also demonstrates that the coupled evolutionary game between the two populations has an invariant set constraint, meaning that under the initial conditions, the constraints are always satisfied during the evolution of the game. The multi-agent formation control problem is transformed into an evolutionary game problem, thus breaking down the centralized optimization problem into several sub-problems, which are then assigned to each sub-agent for solution. Each agent uses its own information, local model, and available neighbor information to solve the sub-problems, significantly reducing computational load and complexity. Furthermore, it overcomes the performance degradation problem caused by insufficient information interaction in traditional decentralized control, maintaining high control performance while improving system flexibility and scalability. This invention employs a leaderless formation control algorithm, where all agents have the same role and function, thus overcoming the shortcomings of the leader-follower formation control algorithm.

[0028] 2. This invention utilizes model predictive control algorithms to construct a global optimization problem. It achieves the formation objective by designing a formation error function within the global model prediction cost function. Furthermore, it introduces the property of invariant sets to ensure that each agent does not collide during movement.

[0029] 3. This invention is also applicable to time-varying communication networks. While improving control and safety performance, it reduces computational complexity and communication burden, solving the problem that some existing formation control algorithms cannot handle communication-constrained or time-varying communication networks. Attached Figure Description

[0030] Figure 1 This is a transformation diagram between the formation control problem and the evolutionary game problem in this invention;

[0031] Figure 2 This is a two-dimensional actual trajectory diagram of the six intelligent agents in this invention;

[0032] Figure 3 This is a graph showing the position coordinates of each agent in this invention over time.

[0033] Figure 4 This is the safe distance-time curve for each agent pair in this invention;

[0034] Figure 5 This is a control input-time curve for each intelligent agent in this invention. Detailed Implementation

[0035] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0036] This invention introduces an evolutionary game theory algorithm into multi-agent formation control. Evolutionary game theory, as a mathematical tool, can describe the behavior of decision-makers when only partial information about some participants is known. Through continuous iteration and optimization, the local behaviors of the participants can achieve a global goal. Therefore, evolutionary game theory is suitable for solving distributed multi-agent formation control problems. This invention provides a distributed model prediction method for leaderless formation control based on distributed evolutionary game theory, comprising the following steps:

[0037] The first part, constructing a multi-agent system, includes the following sub-steps:

[0038] Step 11, System Architecture Design

[0039] Consider a having Formation of multiple agents, taking This represents the position information of the i-th agent. This represents the velocity information of the i-th agent. For any agent... Its dynamic model expression is as follows

[0040]

[0041] in, Let i represent the state variable of the i-th agent. This represents the control variable for the i-th agent.

[0042] Step 12: Determine the communication topology and target for each agent. The communication range of each multi-agent is... Its time-varying communication topology is Here is the node set The corresponding intelligent agent set Vertex set A(k) represents a pair of intelligent agents that can exchange information, A(k) = [a ij (k)] M×M Let a represent the adjacency matrix, where a is the adjacency matrix when agents i and j can exchange information. ij (k) = 1, otherwise a ij (k) = 0. Let Let i represent the desired state of agent i. For any agents i and j, the following must be satisfied:

[0043] (1) Control objective:

[0044] (2) Obstacle avoidance constraint: d ij (k)=||c i (k)-c j (k)||≥R, where the minimum safe distance

[0045] (3) Position constraints: in It is the area that the intelligent agent is allowed to reach;

[0046] (4) Input constraints: in It controls the allowed range of input;

[0047] (5) Expected state requirements: for all That is, the expected target location distance between different intelligent agents is greater than the safe distance; Let i represent the set of neighboring smart agents.

[0048] Step 13: Design a safe distance set for each agent. At each time k, obtain the positions c of agent i and all its neighboring agents j. i (k) and c j (k) uses a Voronoi diagram to reconstruct the constraint set.

[0049]

[0050] in

[0051]

[0052]

[0053]

[0054] Where, δ ij (k), ε ij (k) and ω ij (k) represents the intermediate variables used in the calculation, set It is a polyhedral closed set, i.e., a collision-free set, and for any and All will satisfy ||c i (k)-c j (k)‖≥R.

[0055] Step 14, construct the model prediction optimization problem. To achieve the control objective, let... The cost function is defined as: representing the positional deviation of agent i.

[0056]

[0057] in, and Both are symmetric positive definite matrices, H p For prediction in the time domain, the optimal control problem for UAV formations is described as follows:

[0058] min u(k) J(k) (8a)

[0059] stform = 0, 1, ..., H p -1 (8b)

[0060]

[0061]

[0062]

[0063]

[0064] When the optimization problem (8) has a feasible solution, the optimal control input for a future time period is obtained. Considering the model mismatch and interference in practical applications, the optimal control sequence is not applied to the system one by one. Instead, the first element of the optimal control sequence is used in the actual system. At the next time k+1, the current state of the system is resampled, the optimization problem (8) is reconstructed and solved, and the above steps are repeated. However, the optimization problem constructed at this time is still a centralized optimization problem. In the following steps, the above optimization problem will be solved in a distributed manner using the method of distributed evolutionary game.

[0065] Since collision avoidance constraints are inherently non-convex, they can lead to non-convex optimization problems. To address this computational issue, the idea of ​​Voronoi diagrams is introduced to create a safe distance set for each agent. This guarantees that each agent will not collide as long as it moves within the specified safe distance set.

[0066] Part Two: Two-Population Evolutionary Game under Coupling Constraints. Construct two populations p∈(1,2), each with a large and finite number of participants, and both populations have the same policy set S. Let s i Let S represent the i-th policy, and let m represent the policy set containing n policies. p,i Let represent the number of individuals in population p who accept policy i, and Let ρ be the proportion of population p that accepts strategy i. p,i =m p,i / m p ≥0, and p can be obtained p =[ρ p,1 ,ρ p,2 ,…,ρ p,n ] Tand π p =∑ i∈S ρ p,i =1. Meanwhile, let F be the fitness function of population p. p (p p )=[f p,1 (p p ),f p,2 (p p ),…,f p,n (p p )] T Here, x is uniformly defined. i :=ρ 1,i y i :=ρ 2,i x:=p1, y:=p2, f i x :=f 1,i (p1), f i y :=f 2,i (p2), and

[0067] Step 21, Setting up the communication topology in evolutionary game theory. For two populations (x, y), to maintain a certain balance, the set Ξ={(x, y)∣Ax+By≤C} needs to be satisfied, where A=diag{a1,a2,…,a… n}, B = diag{b1,b2,…,b n} and C = [c1 c2 … c n ] T In the process of evolution, the set Λ:={(x,y)∣∑ i∈S x i =π1,∑ i∈S y i =π2,x i ≥0,y i The state ≥0} contains all possible states of the population. For the first population, the policy interactions between individuals can be represented by an undirected graph. To represent, where the node set Represents all policy sets, vertex sets This represents the different strategies that individuals in population x can adopt, A(k) = [a ij (k)] M×M Let a represent the adjacency matrix, where a is the adjacency matrix when an individual adopts policy i and can also adopt policy j. ij (k) = 1, otherwise a ij (k) = 0. Similarly, for the second population, the policy interactions between individuals can be represented by an undirected graph. To express.

[0068] The optimization problem of evolutionary game theory is solved by finding the Nash equilibrium point, and can be described as follows:

[0069] max x,y W(x,y) (9a)

[0070] stAx+By≤C (9b)

[0071]

[0072]

[0073] x i ≥0 (9e)

[0074] y i ≥0 (9f)

[0075] Wherein, the cost function W(x,y) is a strictly continuous and differentiable concave function, (x i ,y i () represents the population state.

[0076] The evolutionary process of the proportion of populations x and y adopting strategy i can be described by distributed evolutionary dynamics, and its expression is:

[0077]

[0078]

[0079] This dynamic is also called average dynamics. Furthermore, the correction protocol φ ij The current gains and aggregated behavior are taken as inputs, and the transition frequency is output, which is the frequency at which an individual switches from strategy i to strategy j based on the current overall state and gains.

[0080] Step 21, setting the communication protocol. For any given x and y, use... Let S represent a set of ternaries, and for any q∈S, satisfy... but These are the coefficients corresponding to the smallest element of the vector C-(Ax+By). Therefore, the correction protocol for population p can be designed as follows:

[0081]

[0082] Substituting (12) into (10) and (11), we can obtain

[0083]

[0084]

[0085] This is the Distributed Smith Dynamics (DSD2PC) of two populations with coupling constraints, and the evolutionary game with this dynamics is called the Distributed Evolutionary Game with Coupling Constraints (DEG2PC).

[0086] make Then (13) and (14) can be restated as follows:

[0087]

[0088]

[0089] Evolutionary dynamics can be expressed in compact set form as follows:

[0090]

[0091] in, and These are about the diagram. and The Laplace matrix.

[0092] S10. Prove that evolutionary games constrained by two groups possess the property of invariant sets. Given (x,y)∈Ξ∩Λ, by... get And make get

[0093]

[0094]

[0095] Therefore, r x (i,j)=r x (j,i)≥0. Adjacency matrix It can be represented as:

[0096]

[0097] Relationships from the Laplace matrix It can be obtained

[0098]

[0099] According to r x The nonnegativity of (i,j) and the Laplace matrix It is positive semidefinite. The same logic can be used to prove it. It is semi-positive definite and

[0100] S11, according to Lemma 1 and It can be obtained and In other words and It is a constant. Furthermore, when x... i =0 or y i When = 0, according to (13) and (14), we get

[0101]

[0102]

[0103] Therefore, for x i ≥0 and y i ≥0, (x(t),y(t))∈Λ.

[0104] When (x(0), y(0)) ∈ Ξ, once the trajectory (x(t), y(t)) reaches the boundary of set Ξ, for i ∈ S, a i x i +b i y i =c i According to Theorem 1, and a i and b i Substituting (13) and (14), we get

[0105]

[0106]

[0107] Discuss in the following four situations and

[0108] If a i >0,b i >0

[0109] If a i >0,b i ≤0

[0110] If a i ≤0,b i >0

[0111] If a i ≤0,b i ≤0

[0112] Then it always satisfies nonnegativity Non-growth ai x i +b i y i ≤c i Due to the continuity of the trajectory (x,y), it follows that (x(t),y(t))∈Λ in all subsequent time steps. Therefore, the set Ξ∩Λ is an invariant set.

[0113] S12. Choose E(x,y):=W(x * ,y * As a Lyapunov function, and given E(x,y)≥0, its derivative can be expressed as:

[0114]

[0115] Therefore, when the initial value (x(0), y(0))∈Ξ evolves along (13) and (14), DEG2PC approaches the Nash equilibrium point, and the Nash equilibrium point is locally asymptotically stable.

[0116] Part Three: Distributed Model Predictive Control Algorithm Based on DEG2PC Theory:

[0117] Step 31, the transformation diagram between the formation control problem and the evolutionary game problem in this invention is as follows: Figure 1 As shown, using evolutionary game theory, the population state (x) in DEG2PC theory is transformed. i ,y i and position components in optimal control problems Related, the relationship is as follows

[0118]

[0119]

[0120] According to the dynamic model in (6), u i (k+m|k) and v i (k+m+1|k) can be rewritten as:

[0121] u i (k+m|k)=c i (k+m+1|k)-2c i (k+m|k)+c i (k+m-1|k) (19)

[0122] v i (k+m+1|k)=c i (k+m+1|k)-c i (k+m|k) (20)

[0123] Substitute (19) and (20) into the optimization problem (8). Since problem (8) minimizes the cost function J(k) and problem (9) maximizes the concave function W(x,y), the fitness function for each policy can be described as f x =- x J and f y =- y J. Furthermore, constraints (8d), (8e), and (8f) in problem (8) can be transformed into The form corresponds to constraint (9b) in problem (9).

[0124] Step 32: For populations x and y, select the modified protocol as in (12), and use the dynamic evolution of (15) and (16). The population result will tend towards the Nash equilibrium point. Then, at time k, the optimal position trajectory (x) can be obtained. * (k),y * (k) and the optimal control input sequence u * (k). Therefore, the formation control problem was solved in a distributed manner using DEG2PC (9).

[0125] In summary, the distributed model prediction method for leaderless formation control based on distributed evolutionary game theory can be described as follows: Given input: desired position Predicting time domain H p Safety distance R, communication range θ, weight matrix Q i P i and R i Required output: (x * (k),y * (k)) and u i * (k|k)

[0126] (1) Given sample z at time k i (k) and communication topology

[0127] (2) Constructing the formation control problem (8); Selecting Design modification protocol (12);

[0128] (3) For each strategy f x and f y Obtain the appropriate function;

[0129] (4) Solve for the optimal position trajectory (x) using (13) and (14). * (k),y * (k) and the optimal control input sequence u * (k);

[0130] (5) will u i * Substitute (k|k) into each agent and repeat the above operation.

[0131] Part Four: Theoretical Simulation. A multi-agent system with six agents is selected. For each agent... The system model is

[0132]

[0133] Input constraints for each agent The communication range θ = 2.3, the safety distance R = 0.5, and the prediction time domain H p =20, weight matrix Q i =R i =P i =I 4×4 Each agent's initial velocity and desired velocity are set to 0, and its initial position is...

[0134] c1(0) = [3 3] T c2(0) = [1 4] T c3(0) = [2 0] T

[0135] c4(0) = [4 1] T c5(0) = [0 2] T c6(0) = [3 5] T

[0136] In order to form a formation, the desired position of each agent is:

[0137]

[0138]

[0139] Simulation experiments were conducted using the ICLOCS and PDToolbox solver tools in MATLAB. The results are shown in the attached figures. Figure 2 This is a two-dimensional actual trajectory diagram of the six intelligent agents in this invention. Figure 3 This is a graph showing the position coordinates of each agent in this invention versus time. Figure 4 This is the safe distance-time curve for each agent pair in this invention. Figure 5 This is a control input-time curve for each intelligent agent in this invention. Figure 2 Simulation results show that, under the control of this algorithm, each agent can eventually reach the designated target point. Figure 3 This refers to the position of each agent during its movement. Figure 4The subgraphs in the diagram show the relative positions between the agents. It can be seen that the relative positions between the agents are always greater than the safe distance of 0.5, which means that the agents have the effect of collision avoidance. Figure 4 The results show that the agent can ensure that the input constraints are satisfied during the movement process.

[0140] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A model-based prediction method for leaderless formation control based on distributed evolutionary game theory, characterized in that, Includes the following steps: Step 1: Establish a multi-agent system, define the initial and target positions of the agents, construct a dynamic model of the agents, and solve the optimal control problem involving obstacle avoidance constraints among the agents, control constraints of the agents, and state constraints; wherein, in order to achieve the control objective, let The cost function is defined as: representing the positional deviation of agent i. in, and Both are symmetric positive definite matrices, H p For prediction in the time domain, the optimal control problem for UAV formations is described as follows: min u(k) J(k) s.t.form=0,1,…,H p -1 Where: J(k) is the defined cost function, This represents the position information of the i-th agent. This represents the speed information of the i-th agent. Let i represent the state variable of the i-th agent. This represents the control variable for the i-th agent. This represents the collision avoidance constraint set for the i-th agent. Indicates the movable range of multiple agents. H represents the allowed control output range of a single agent. p For prediction in the time domain; The optimal control problem is to predict the state of the agent over a future period of time using a prediction model, given that the final target state is known, so as to minimize the distance between the agent's position and the target position over the future period of time and obtain the optimal control input at the current moment. Step 2: Create a safe distance set for each agent, ensuring that each agent will not collide as long as it moves within the specified safe distance set; the safe distance set is defined as follows: Where R is the specified safety distance, set It is a polyhedral closed set, for any and Satisfy‖c i (k)-c j (k)‖≥R, Let δ represent the set of neighboring agents of agent i. ij (k), ε ij (k) and ω ij (k) represents the intermediate variable used in the calculation; Step 3: Propose two group evolutionary games under coupling constraints, select a modification protocol to construct evolutionary dynamic equations, so that the evolutionary dynamic equations of each population can reach the Nash equilibrium solution of the game through continuous iteration and optimization, and have the property of invariant set. Specifically, two populations p∈(1,2) are constructed, each with a large and finite number of participants, and both populations have the same policy set S; let s i ∈S, s i Let m represent the i-th policy, S represent the policy set containing n policies, and m represent the policy set containing n policies. p,i Let represent the number of individuals in population p who accept policy i, and Let ρ be the proportion of population p that accepts strategy i. p,i =m p,i / m p ≥0, we get p p =[ρ p,1 ,ρ p,2 ,…,ρ p,n ] T and π p =∑ i∈S ρ p,i =1, and let the fitness function of population p be F. p (p p )=[f p,1 (p p ),f p,2 (p p ),…,f p,n (p p )] T ; Unified definition of x i :=ρ 1,i y i :=ρ 2,i x:=p1, y:=p2, f i x :=f 1,i (p1), f i y :=f 2,i (p2), and In the communication topology setting of evolutionary game theory, for two populations (x, y), the set Ξ={(x, y)∣Ax+By≤C} is satisfied, where A=diag{a1,a2,…,a… n }, B = diag{b1,b2,…,b n } and C = [c1 c2…c n ] T In the process of evolution, the set Λ:={(x,y)∣∑ i∈S x i =π1,∑ i∈S y i =π2,x i ≥0,y i ≥0} contains all possible states of the population; for the first population, the policy interactions between individuals are represented by an undirected graph. To represent, where the node set Represents all policy sets, vertex sets This represents the different strategies that individuals in population x can adopt, A(k) = [a ij (k)] M×M Let a represent the adjacency matrix, where a is the adjacency matrix when an individual adopts policy i and can also adopt policy j. ij (k) = 1, otherwise a ij (k) = 0; similarly, for the second population, the policy interactions between individuals are represented by an undirected graph. To indicate; The optimization problem of evolutionary game theory is solved by finding the Nash equilibrium point, specifically as follows: max x,y W(x,y) stAx+By≤C x i ≥0 y i ≥0 Wherein, the cost function W(x,y) is a strictly continuous and differentiable concave function, (x i ,y i ) represents the population state; The evolutionary process of the proportion of populations x and y adopting strategy i is described by distributed evolutionary dynamics, and its expression is: Amendment Protocol φ ij The current gains and aggregated behavior are taken as inputs, and the transition frequency is output, which is the frequency at which an individual switches from strategy i to strategy j based on the current overall state and gains. In the communication protocol settings, for any given x and y, use Let S represent a set of ternaries, and for any q∈S, satisfy... but These are the coefficients corresponding to the smallest element of the vector C-(Ax+By); the correction protocol for population p is designed as follows: And thus obtain Step 4: Transform the constructed multi-agent formation problem into two group evolutionary game problems with coupling constraints, and use the evolutionary dynamics equations of the evolutionary game to solve the multi-agent formation optimization problem. Specifically, using evolutionary game theory, the population state (x) in DEG2PC theory is transformed into... i ,y i and position components in optimal control problems Related, the relationship is as follows According to the dynamic model, u i (k+m|k) and v i (k+m+1|k) can be rewritten as: at i (k+m|k)=c i (k+m+1|k)-2c i (k+m|k)+c i (k+m-1|k) v i (k+m+1|k)=c i (k+m+1|k)-c i (k+m|k) Substituting this into the optimal control problem, the fitness function for each strategy is described as f. x =- x J and f y =- y J; The constraints in the optimal control problem are transformed into The form corresponds to the constraints in the expression of the optimization problem of evolutionary game by finding the Nash equilibrium point; For populations x and y, a correction protocol is chosen for population p, using dynamic evolution. After the population outcome tends towards the Nash equilibrium point, the optimal position trajectory (x) is obtained at time k. * (k),y * (k) and the optimal control input sequence u * (k).

2. The method according to claim 1, characterized in that, In step 4, the positions of agents in formation control are transformed into population states in evolutionary game, and each agent in formation control is transformed into a strategy in evolutionary game. The cost function in formation control problem and the benefit function in evolutionary game are combined, and then the optimal control problem in step 1 is solved using evolutionary dynamics equations.

3. The method according to claim 1 or 2, characterized in that, In step 3, a distributed evolutionary game with two populations having coupling constraints is used. The specific steps are as follows: the optimization problem of the evolutionary game is solved by finding the Nash equilibrium point; the optimization problem of finding the Nash equilibrium point is substituted into the average dynamics to obtain the distributed Mish dynamics equations of the two populations with coupling constraints.

Images