Differential game-based fault-tolerant formation encircling control method for fixed-wing unmanned aerial vehicles

Abstract

The invention discloses a fixed-wing unmanned aerial vehicle fault-tolerant formation encircling control method based on differential game, and the method comprises the steps: firstly building a mathematical model of leader and follower unmanned aerial vehicles, an actuator fault model, and an information interaction topological graph of a multi-unmanned aerial vehicle system; secondly, under a differential game framework, designing a formation tracking control performance index of the leader unmanned aerial vehicle and a surrounding control performance index of the follower unmanned aerial vehicle; then, designing an optimal fault-tolerant game control strategy model of each unmanned aerial vehicle and resolving a corresponding optimal fault-tolerant control strategy; and finally, proposing an optimal fault-tolerant control scheme based on a self-adaptive dynamic planning technology of the evaluation neural network, and controlling the fixed-wing unmanned aerial vehicle. The state of the unmanned aerial vehicle system can be asymptotically stable, so that the system can complete a formation tracking and encircling control task under the condition of an actuator fault.

Description

Technical Field

[0001] The invention relates to fault-tolerant control and fault-tolerant formation encirclement control of fixed-wing unmanned aerial vehicles, and in particular to a fault-tolerant formation encirclement control method for fixed-wing unmanned aerial vehicles based on differential games. Background Art

[0002] As an important research object of multi-UAV systems, cooperative control has attracted the attention of many scholars in the fields of consistency control, formation control, encirclement control and siege control. However, basic cooperative control technology is difficult to handle some complex engineering problems, such as multi-UAV escort and target tracking. Therefore, by combining the advantages of formation tracking control and siege control, a formation siege control is proposed, which enables the leader UAV to track the target while forming the desired formation and the follower UAVs to converge to the convex hull formed by all leader UAVs.

[0003] Faced with complex external environments and the increasing number of drones, drone systems are bound to fail. Therefore, ensuring the reliability and stability of drones is a key step. Fault-tolerant control allows the system to maintain stability and expected performance when a fault occurs. At present, fault-tolerant control methods are widely used in formation tracking control and encirclement control problems of faulty multi-drone systems. However, for the formation tracking and encirclement control problems of faulty multi-drone systems, physical faults such as aging and jamming cause the formation tracking error and encirclement error to fail to achieve the control target at the same time, which increases the difficulty of controller design and there are few related research results.

[0004] In addition, in order to achieve the stability and optimality of the faulty multi-UAV system at the same time, we introduced the differential game method. Differential games mainly focus on dynamic decision-making in continuous time, which can essentially be regarded as the optimal control problem of two or more players. When dealing with nonlinear systems, it usually involves solving the Hamilton-Jacobi-Isaacs (HJI) equation, but due to the high nonlinearity of the HJI equation, it is difficult to obtain its analytical solution. To this end, adaptive dynamic programming technology provides a novel solution method, which implements the approximate mechanism of the HJI equation by constructing a neural network, thereby obtaining the approximate optimal control strategy of the system. In addition, adaptive dynamic programming technology has good convergence and stability, and is more suitable for flight control. Therefore, the present invention uses adaptive dynamic programming technology to analyze the formation tracking and encirclement control problem of faulty fixed-wing UAVs under the framework of differential games. Summary of the invention

[0005] Purpose of the invention: The present invention proposes a fault-tolerant formation encirclement control method for fixed-wing UAVs based on differential game, which solves the fault-tolerant formation encirclement control problem of multiple fixed-wing UAVs with complex dynamics and overcomes the influence of actuator failure on the control performance of the UAV system.

[0006] Technical solution: The present invention discloses a method for controlling a fixed-wing UAV formation based on differential game, which comprises the following steps:

[0007] (1) Establish mathematical models of leader and follower UAVs and actuator failure models of follower UAVs;

[0008] (2) According to the requirements of formation encirclement control, the formation tracking error dynamics model of the leader UAV and the encirclement error dynamics model of the follower UAV are established;

[0009] (3) In the framework of differential games, a leader UAV formation control problem based on graph games is designed, and the distributed optimal control strategy of the i-th leader UAV is obtained;

[0010] (4) In the framework of differential games, a zero-sum game-based fault-tolerant encirclement control problem for follower UAVs is designed, and the optimal fault-tolerant control strategy and the most serious actuator bias fault of the kth follower UAV are obtained;

[0011] (5) Based on the adaptive dynamic programming technology of critical neural network, an optimal fault-tolerant control scheme is proposed to control the fixed-wing UAV.

[0012] Furthermore, the mathematical model of the leader UAV in step (1) is:

[0013] The multi-fixed-wing UAV system consists of m leader UAVs and (nm) follower UAVs. The set of all leader UAVs is defined as The set of all follower drones is defined as The power model of the leader drone is:

[0014]

[0015] Where i∈L,q i =[x i ,y i ,z i ] T is the position vector of the i-th leader UAV; i , and γ i They represent the speed, yaw angle and pitch angle of the i-th leader UAV respectively; T i , n i and φ i Denote the thrust, lift coefficient and roll angle of the i-th leader UAV respectively; D i is the aerodynamic resistance, m i is the mass of the i-th leader drone, g is the gravitational acceleration;

[0016] The power model of the leader UAV is derived, and then the affine nonlinear system model of the leader is constructed as follows:

[0017]

[0018] in, μ i is the virtual input signal,

[0019]

[0020] Furthermore, the mathematical model of the follower drone in step (1) is:

[0021] The dynamic model of the follower UAV is expressed as:

[0022]

[0023] The dynamic model of the follower UAV is derived, and the affine nonlinear system model of the follower is established as follows:

[0024]

[0025] in,

[0026] Furthermore, the actuator failure model of the follower UAV in step (1) is:

[0027]

[0028] Among them, k∈F, F is defined as the set of all follower drones, is the control signal of the application, is the designed control signal, Indicates the bias fault occurred in the engine, elevator, aileron and rudder of the kth follower UAV.

[0029] Furthermore, the implementation process of step (2) is as follows:

[0030] The local position error and local velocity error of the i-th leader UAV are:

[0031]

[0032] Where i∈L, b i is the traction control coefficient between the i-th leader UAV and the target; let The formation tracking error dynamics model of the i-th leader UAV is:

[0033]

[0034] in,

[0035] The encirclement error of the kth follower UAV is:

[0036]

[0037] make The encirclement error dynamics model of the kth follower UAV is obtained as:

[0038]

[0039] in,

[0040] Furthermore, the implementation process of step (3) is as follows:

[0041] Based on the formation error, flight cost and terminal moment formation error generated during the flight of the leader UAV, the following cost function is constructed:

[0042]

[0043] in, Weight matrix Q i >0, Q if >0, R ii >0, R ij ≥0;

[0044] When the optimal control strategy When exists, the optimal value function and optimal cost function of the design graph game are:

[0045]

[0046]

[0047] In the graph game of the leader drone, each drone strives to minimize its cost function Finally, we get an optimal control strategy that satisfies Nash equilibrium.

[0048] Construct a cost function J i Formation tracking error dynamics The Hamiltonian function is:

[0049]

[0050] Among them, the value function V i Satisfies the HJI equation The distributed optimal control strategy of the i-th leader UAV is obtained as:

[0051]

[0052] Furthermore, the implementation process of step (4) is as follows:

[0053] The control strategy μ k and actuator fault f k As opposing participants in the game, the fault-tolerant encirclement control problem of follower drones based on zero-sum games is studied, and the following cost function is established:

[0054]

[0055] in, Weight matrix K k >0,K kf >0,P kk >0,P ks ≥0, T kk >0, T ks ≥0, constant θ>0;

[0056] When the saddle point When exists, the optimal value function and optimal cost function of the zero-sum game are constructed as:

[0057]

[0058] In the zero-sum game of follower drones, each drone strives to minimize its cost function Finally, the optimal fault-tolerant control strategy that satisfies Nash equilibrium is obtained. and the most severe actuator failure

[0059] Construct a cost function J k Enclosure Error Dynamics The Hamiltonian function is:

[0060]

[0061] Among them, the value function V k Satisfies the HJI equation V k (0) = 0; the optimal fault-tolerant control strategy and the most serious actuator bias fault of the kth follower UAV are obtained:

[0062]

[0063] Furthermore, the implementation process of step (5) is as follows:

[0064] According to Weierstrass's high-order approximation theorem, the approximate optimal value function of the leader and follower drones based on the critical neural network is constructed:

[0065]

[0066] Among them, W i ,W k ∈R l is the ideal neural network weight vector, σ i (α i ),σ k (β k )∈R l is the activation function vector, l is the number of neurons in the hidden layer of the critical neural network, ε i (α i ),ε k (β k ) is the estimation error of the neural network; the value function V i * (α i )and To α i and β k The derivatives are:

[0067]

[0068] The optimal control strategy of the leader UAV, the optimal fault-tolerant control strategy of the follower UAV, and the worst actuator failure are obtained as follows:

[0069]

[0070] Correspondingly, the Hamiltonian functions of the leader and follower drones are expressed as:

[0071]

[0072] According to Weierstrass theorem, when l→∞, the neural network estimation error ε i (α i ),ε k (β k ) tends to 0; in order to derive the unknown ideal weight W i ,W k , the approximate value function of constructing the leader and follower drones is:

[0073]

[0074] in, are the ideal weight vectors W i , W kThe approximate value of , the weight estimation error is Taking partial derivatives of the above approximate value function yields:

[0075]

[0076] The approximate Nash equilibrium solution of the multi-UAV system is obtained:

[0077]

[0078] Accordingly, the approximate Hamiltonian functions of the leader and follower drones are obtained:

[0079]

[0080] Furthermore, the estimated error of the Hamiltonian function is:

[0081]

[0082] in,

[0083]

[0084]

[0085] In addition, assuming that i and k The norm is bounded, that is, ||ζ i ||≤ζ Mi ,||ζ k ||≤ζ Mk , Mi and Mk are greater than 0; in order to minimize the estimation error Ξ i ,Ξ k , we need to minimize the squared residual and To update the neural network weights Therefore, the gradient descent method is used to design the following cooperative regulation law of the leader and follower drones:

[0086]

[0087] Among them, l i ,l k >0 is the learning rate of the critical neural network. is an adjustable parameter, They are Mi ,ζ Mk The estimated value of is, and the estimated error is

[0088] Further get the neural network weights of the leader and follower drones The approximate error is:

[0089]

[0090] Beneficial effects: Compared with the prior art, the beneficial effects of the present invention are as follows: Based on the differential game method, the present invention solves the fault-tolerant formation encirclement control problem of multiple fixed-wing UAVs with complex dynamics, overcomes the influence of actuator failure on the control performance of the UAV system, and simultaneously achieves the stability and optimal performance of multiple fixed-wing UAVs; the adaptive neural network coordination law designed by the present invention can make the neural network weight estimation error and the state of the multi-UAV system asymptotically stable. Compared with the ultimately uniformly bounded results in the prior art, when the UAV performs tasks with high precision requirements, the asymptotic stability of the system state can reduce the difficulty of execution. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 is a flow chart of the present invention;

[0092] Figure 2 A schematic diagram for describing the fixed-wing UAV formation tracking and encirclement control mission;

[0093] Figure 3 The schematic diagram of the communication topology of five fixed-wing UAVs;

[0094] Figure 4 This is a schematic diagram of the flight paths of five fixed-wing drones;

[0095] Figure 5 Schematic diagram of the formation tracking error of three leader drones;

[0096] Figure 6 Schematic diagram of the local velocity errors of the three leader UAVs;

[0097] Figure 7 Schematic diagram of the encirclement error of two follower drones;

[0098] Figure 8 Schematic diagram of the local velocity error of two follower UAVs;

[0099] Fig. 9 Schematic diagram of the control input signals for the leader drone;

[0100] Fig.10 Schematic diagram of the control input signal for the follower drone; Fig.11 Schematic diagram of the neural network weights critiqued for the first drone. DETAILED DESCRIPTION

[0101] The present invention is further described in detail below with reference to the accompanying drawings.

[0102] like Figure 1 As shown, the present invention proposes a fixed-wing UAV fault-tolerant formation encirclement control method based on differential game, which specifically includes the following steps:

[0103] Step 1, establish mathematical models of leader and follower drones and actuator failure models.

[0104] The multi-fixed-wing UAV system consists of m leader UAVs and nm follower UAVs. The set of all leader UAVs is defined as The set of all follower drones is defined as Among them, the power model of the leader drone is:

[0105]

[0106] Where i∈L,q i =[x i ,y i ,z i ] T is the position vector of the i-th leader UAV. i , and γ i T denotes the speed, yaw angle and pitch angle of the i-th leader UAV respectively. i , n i and φ i Denote the thrust, lift coefficient and roll angle of the i-th leader UAV respectively. i is the aerodynamic resistance, m i is the mass of the i-th leader UAV, and g is the gravitational acceleration.

[0107] The power model of the leader UAV is derived, and then the affine nonlinear system model of the leader is constructed as follows:

[0108]

[0109] in, μ i is the virtual input signal,

[0110]

[0111] Considering the actuator bias failure of the follower UAV, the fault model is established:

[0112]

[0113] Among them, k∈F, is the control signal of the application, is the designed control signal, Indicates the bias fault occurred in the engine, elevator, aileron and rudder of the kth follower UAV.

[0114] Therefore, the dynamic model of the follower UAV can be expressed as:

[0115]

[0116] Similarly, the dynamic model of the follower UAV is derived to establish the affine nonlinear system model of the follower:

[0117]

[0118] in,

[0119] Step 2: Establish the formation tracking error dynamics model of the leader UAV and the encirclement error dynamics model of the follower UAV.

[0120] The local position error and local velocity error of the i-th leader UAV (i∈L) are:

[0121]

[0122] Among them, b i is the traction control coefficient between the i-th leader UAV and the target.

[0123] make The formation tracking error dynamics model of the i-th leader UAV is obtained as:

[0124]

[0125] in,

[0126] The encirclement error of the kth follower UAV (k∈F) is:

[0127]

[0128] make The encirclement error dynamics model of the kth follower UAV is obtained as:

[0129]

[0130] in,

[0131] The purpose of the present invention is to minimize the local formation error of the leader UAV and the encirclement error of the follower UAV, so as to achieve Figure 2 The formation shown is tracking and encircling the mission.

[0132] Step 3: Under the differential game framework, design the formation control problem of the leader UAV and the fault-tolerant encirclement control problem of the follower UAV.

[0133] Leader UAV formation tracking control problem based on differential graph game: In order to achieve the flight consistency of the leader UAV and optimize its control performance, the formation tracking control problem of the leader UAV is studied under the framework of differential graph game. Considering the formation error, flight cost and formation error at the terminal moment generated during the flight of the leader UAV, the following cost function is constructed:

[0134]

[0135] in, Weight matrix Q i >0,Q if >0, R ii >0, R ij ≥0.

[0136] When the optimal control strategy When exists, the optimal value function and optimal cost function of the design graph game are

[0137]

[0138] In the graph game of the leader drone, each drone strives to minimize its cost function Finally, we get an optimal control strategy that satisfies Nash equilibrium.

[0139] Construct a cost function J i Formation tracking error dynamics The Hamiltonian function is

[0140]

[0141] Among them, the value function V i Satisfies the HJI equation V i (0)=0.

[0142] Therefore, the distributed optimal control strategy of the i-th leader UAV can be obtained as follows:

[0143] Fault-tolerant encirclement control problem of follower drones based on zero-sum game: k and actuator fault f k As opposing participants in the game, the fault-tolerant encirclement control problem of follower drones based on zero-sum games is studied, and the following cost function is established:

[0144]

[0145] in, Weight matrix K k >0,K kf >0,P kk >0,P ks ≥0, T kk >0, T ks ≥0, constant θ>0.

[0146] When the saddle point When exists, the optimal value function and optimal cost function of the zero-sum game are constructed as:

[0147]

[0148] In the zero-sum game of follower drones, each drone strives to minimize its cost function Finally, the optimal fault-tolerant control strategy that satisfies Nash equilibrium is obtained. and the most severe actuator failure

[0149] Construct a cost function J k Enclosure Error Dynamics The Hamiltonian function is:

[0150]

[0151] Among them, the value function V k Satisfies the HJI equation V k (0)=0.

[0152] Therefore, the optimal fault-tolerant control strategy for the kth follower UAV can be obtained: Worst actuator bias fault

[0153] Step 4, based on the adaptive dynamic programming technology of critical neural network, an optimal fault-tolerant control scheme is proposed.

[0154] According to Weierstrass's high-order approximation theorem, the approximate optimal value function of the leader and follower drones based on the critical neural network is constructed:

[0155]

[0156] Among them, W i ,W k ∈R l is the ideal neural network weight vector, σi (α i ),σ k (β k )∈R l is the activation function vector, l is the number of neurons in the hidden layer of the critical neural network, ε i (α i ),ε k (β k ) is the estimation error of the neural network. The value function V i * (α i )and To α i and β k The derivatives are:

[0157]

[0158] The optimal control strategy of the leader UAV, the optimal fault-tolerant control strategy of the follower UAV, and the worst actuator failure are obtained as follows:

[0159]

[0160] Correspondingly, the Hamiltonian functions of the leader and follower drones can be expressed as:

[0161]

[0162] According to Weierstrass theorem, when l→∞, the neural network estimation error ε i (α i ),ε k (β k ) tends to 0. In order to derive the unknown ideal weight W i ,W k , construct the approximate value function of the leader and follower drones as

[0163]

[0164] in, are the ideal weight vectors W i , W k The approximate value of , the weight estimation error is Taking partial derivatives of the above approximate value function yields:

[0165]

[0166] Therefore, the approximate Nash equilibrium solution of the multi-UAV system is obtained.

[0167]

[0168] Accordingly, the approximate Hamiltonian functions of the leader and follower drones are obtained:

[0169]

[0170] Furthermore, the estimated error of the Hamiltonian function is:

[0171]

[0172]

[0173] in,

[0174]

[0175] In addition, assuming that i and k The norm is bounded, that is, ||ζ i ||≤ζ Mi ,||ζ k ||≤ζ Mk , Mi and Mk are all greater than 0. In order to minimize the estimation error Ξ i ,Ξ k , we need to minimize the squared residual and To update the neural network weights Therefore, the gradient descent method is used to design the following cooperative regulation law of the leader and follower drones:

[0176]

[0177] Among them, l i ,l k >0 is the learning rate of the critical neural network. is an adjustable parameter, They are Mi ,ζ Mk The estimated value of is, and the estimated error is

[0178] We can further obtain the neural network weights of the leader and follower drones The approximate error is:

[0179]

[0180] Fixed-wing UAV is controlled using a fault-tolerant game controller optimized by critical neural network learning.

[0181] The present invention conducts simulation verification experiments on the designed optimal fault-tolerant game control method in the MATLAB / SIMULINK environment. Figure 3 As shown in Figure 2, a fixed-wing UAV system consisting of three leader UAVs and two follower UAVs is considered. It is assumed that the fourth UAV has an actuator bias fault f4 = [1.5, 0.6, 0.1] after 2.5s. T The weight of an edge in a directed graph is w ij =w ks =1, the weight matrix of the cost function is Q i =Q if =0.01I6, R ii =R ij =0.01I3,K i =K if =0.01I6,P kk =P ks =0.01I3,T ks =0.001I3.

[0182] The control parameters of the critical neural network are l1=11, l2=5.5, l3=5, l4=0.1, l5=0.5, l ζi = l ζk =1, The initial position of each drone is q1(0) = [5.5, 0.5, -1] T m,q2(0)=[5.5,3,-2] T m,q3(0)=[4,5,-2] T m,q4(0)=[11,3,-2] T m,q5(0)=[9,3,-2] T The initial velocities are v1(0) = 5 m / s, v2(0) = 3 m / s, v3(0) = 2.5 m / s, v4(0) = 3.5 m / s, v5(0) = 3 m / s, the initial yaw angles are γ1(0) = 0.95°, γ2(0) = 0.55°, γ3(0) = 0°, γ4(0) = 0°, γ5(0) = 0.55°, and the initial pitch angles are The target trajectory tracked by the leader drone is q0(t)=[3t+2,2t,0.3t+1] T m. The expected distance between adjacent drones is d 10 =[2,0,1] T m, d 21 =[5,5,0] T m, d 31 =[-5,5,0] T m,d 32 =[-10,0,0] Tm.

[0183] like Figure 4 As shown in Figure 1, the three leader drones form a triangular formation while tracking the target, and the two follower drones converge into the convex hull formed by all leader drones. Figure 5 and Figure 6 As shown in , the local position error and local velocity error of the leader UAV converge to 0, that is, the tracking error of the leader UAV formation is asymptotically stable. Figure 7 and Figure 8 As shown in , the encirclement error of the follower UAV is also asymptotically stable. Fig. 9 and Fig.10 As shown in Figure 2, the control input signal of each UAV tends to a constant value. It is worth noting that at 2.5s, due to the actuator failure of the fourth UAV and the propagation of the failure through the communication link, the multi-UAV system promptly adjusted the control input signals μ4 and μ5 to ensure the stability of the flight process. Fig.11 As shown, after the learning process of the adaptive regulation law, the weight estimation error of the critical neural network of the first UAV converges asymptotically to a constant value.

[0184] From the above, it can be seen that the optimal fault-tolerant game control scheme proposed by the present invention for the fixed-wing UAV system with actuator failure is effective, ensuring that the multi-UAV system can still complete the formation tracking and encirclement control task under the actuator failure within the set threshold.

[0185] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.

Claims

1. A fixed-wing UAV fault-tolerant formation encirclement control method based on differential game, characterized in that: The steps include: (1) Establish mathematical models of leader and follower UAVs and actuator failure models of follower UAVs; (2) According to the requirements of formation encirclement control, the formation tracking error dynamics model of the leader UAV and the encirclement error dynamics model of the follower UAV are established; (3) In the framework of differential games, a leader UAV formation control problem based on graph games is designed, and the distributed optimal control strategy of the i-th leader UAV is obtained; (4) In the framework of differential games, a zero-sum game-based fault-tolerant encirclement control problem for follower UAVs is designed, and the optimal fault-tolerant control strategy and the most serious actuator bias fault of the kth follower UAV are obtained; (5) Based on the adaptive dynamic programming technology of critical neural network, an optimal fault-tolerant control scheme is proposed to control the fixed-wing UAV.

2. According to the differential game-based fixed-wing UAV fault-tolerant formation encirclement control method of claim 1, it is characterized in that: The mathematical model of the leader UAV in step (1) is: The multi-fixed-wing UAV system consists of m leader UAVs and (nm) follower UAVs. The set of all leader UAVs is defined as The set of all follower drones is defined as The power model of the leader drone is: Where i∈L,q i =[x i ,y i ,z i ] T is the position vector of the i-th leader UAV; i , and γ i They represent the speed, yaw angle and pitch angle of the i-th leader UAV respectively; T i , n i and φ i Denote the thrust, lift coefficient and roll angle of the i-th leader UAV respectively; D i is the aerodynamic resistance, m i is the mass of the i-th leader drone, g is the gravitational acceleration; The power model of the leader UAV is derived, and then the affine nonlinear system model of the leader is constructed as follows: in, μ i is the virtual input signal, 3. According to the differential game-based fixed-wing UAV fault-tolerant formation encirclement control method of claim 1, it is characterized in that: The mathematical model of the follower drone in step (1) is: The dynamic model of the follower UAV is expressed as: The dynamic model of the follower UAV is derived and the affine nonlinear system model of the follower is established as follows: in, 4. According to the differential game-based fixed-wing UAV fault-tolerant formation encirclement control method of claim 1, it is characterized in that: The actuator failure model of the follower UAV in step (1) is: Among them, k∈F, F is defined as the set of all follower drones, is the control signal of the application, is the designed control signal, Indicates the bias fault occurred in the engine, elevator, aileron and rudder of the kth follower UAV.

5. The method for controlling fixed-wing UAV formation encirclement based on differential game according to claim 1 is characterized in that: The implementation process of step (2) is as follows: The local position error and local velocity error of the i-th leader UAV are: Where i∈L, b i is the traction control coefficient between the i-th leader UAV and the target; The formation tracking error dynamics model of the i-th leader UAV is: in, The encirclement error of the kth follower UAV is: make The encirclement error dynamics model of the kth follower UAV is: in, 6. The method for controlling fixed-wing UAV formation encirclement based on differential game according to claim 1 is characterized in that: The implementation process of step (3) is as follows: Based on the formation error, flight cost and terminal moment formation error generated during the flight of the leader UAV, the following cost function is constructed: in, Weight matrix Q i >0, Q if >0, R ii >0, R ij ≥0; When the optimal control strategy When exists, the optimal value function and optimal cost function of the design graph game are: In the graph game of the leader drone, each drone strives to minimize its cost function Finally, we get an optimal control strategy that satisfies Nash equilibrium. Construct a cost function J i Formation tracking error dynamics The Hamiltonian function is: Among them, the value function V i Satisfies the HJI equation V i (0) = 0; the distributed optimal control strategy of the i-th leader UAV is obtained as follows:

7. The method for controlling fixed-wing UAV formation encirclement based on differential game according to claim 1 is characterized in that: The implementation process of step (4) is as follows: The control strategy μ k and actuator fault f k As opposing participants in the game, the fault-tolerant encirclement control problem of follower drones based on zero-sum games is studied, and the following cost function is established: in, Weight matrix K k >0,K kf >0,P kk >0,P ks ≥0, T kk >0, T ks ≥0, constant θ>0; When the saddle point When exists, the optimal value function and optimal cost function of the zero-sum game are constructed as: In the zero-sum game of follower drones, each drone strives to minimize its cost function Finally, the optimal fault-tolerant control strategy that satisfies Nash equilibrium is obtained. and the most severe actuator failure Construct a cost function J k Enclosure Error Dynamics The Hamiltonian function is: Among them, the value function V k Satisfies the HJI equation V k (0) = 0; the optimal fault-tolerant control strategy and the most serious actuator bias fault of the kth follower UAV are obtained:

8. The method for controlling fixed-wing UAV fault-tolerant formation encirclement based on differential game according to claim 1, characterized in that: The implementation process of step (5) is as follows: According to Weierstrass's high-order approximation theorem, the approximate optimal value function of the leader and follower drones based on the critical neural network is constructed: Among them, W i ,W k ∈R l is the ideal neural network weight vector, σ i (α i ),σ k (β k )∈R l is the activation function vector, l is the number of neurons in the hidden layer of the critical neural network, ε i (α i ),ε k (β k ) is the estimation error of the neural network; the value function and To α i and β k The derivatives of are: The optimal control strategy of the leader UAV, the optimal fault-tolerant control strategy of the follower UAV, and the worst actuator failure are obtained as follows: Correspondingly, the Hamiltonian functions of the leader and follower drones are expressed as: According to Weierstrass theorem, when l→∞, the neural network estimation error ε i (α i ),ε k (β k ) tends to 0; in order to derive the unknown ideal weight W i ,W k , the approximate value function of constructing the leader and follower drones is: in, are the ideal weight vectors W i , W k The approximate value of , the weight estimation error is Taking partial derivatives of the above approximate value function yields: The approximate Nash equilibrium solution of the multi-UAV system is obtained: Accordingly, the approximate Hamiltonian functions of the leader and follower drones are obtained: Furthermore, the estimated error of the Hamiltonian function is: in, In addition, assuming that i and k The norm is bounded, that is, ||ζ i ||≤ζ Mi ,||ζ k ||≤ζ Mk , Mi and Mk are greater than 0; in order to minimize the estimation error Ξ i ,Ξ k , we need to minimize the squared residual and To update the neural network weights Therefore, the gradient descent method is used to design the following cooperative regulation law of the leader and follower drones: Among them, l i ,l k >0 is the learning rate of the critical neural network. is an adjustable parameter, They are Mi ,ζ Mk The estimated value of is, and the estimated error is Further get the neural network weights of the leader and follower drones The approximate error is:

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