A fault-tolerant game control method for quadrotor helicopter cluster formation

Abstract

This invention discloses a fault-tolerant game-theoretic control method for quadrotor helicopter swarm formations. It designs a quadrotor helicopter swarm formation mode, constructs an information interaction topology graph of the helicopter swarm, and establishes fault models for the rotating and translational subsystems. An adaptive fault diagnosis observer is designed to estimate actuator faults. Using the translational subsystem as the leader and the rotating subsystem as the follower, an inner and outer loop fault-tolerant game framework based on an incentive-based Steinberg differential graph game is constructed. Through full-drive system theory, fault-tolerant game controllers for the rotating and translational subsystems are designed separately, and a linear closed-loop system of the quadrotor helicopters is constructed. Performance indices for the rotating and translational subsystems are designed, and an optimal fault-tolerant game-theoretic control strategy is designed based on differential game theory. This invention enables stable formation control of quadrotor helicopter swarms, effectively improves the fault tolerance of helicopter swarms, and enhances the stability and intelligence of the helicopter swarm system.

Description

Technical Field

[0001] This invention belongs to the technical field of multi-aircraft control, specifically relating to a fault-tolerant game-theoretic control method for quadcopter helicopter swarm formations. Background Technology

[0002] Quadrotor helicopters are widely used in both civilian and military fields due to their flexible maneuverability and broad application scenarios, such as forest fire monitoring, rescue missions, and logistics delivery. To achieve efficient and stable swarm formation control, game theory, as a powerful analytical tool, has been introduced into helicopter swarm control strategies. For example, in multi-agent systems, game theory, by studying the cooperative and competitive relationships among agents, can accomplish tasks such as optimizing formation control and maintaining formation.

[0003] Existing research on quadrotor helicopter swarm control largely focuses on the influence of the outer loop (translational subsystem) on the inner loop (rotational subsystem), neglecting the bidirectional interaction between the inner and outer loops. This results in low efficiency and coordination of the control system under complex tasks and highly dynamic environments. Furthermore, existing control strategies typically employ complex nonlinear solution methods, such as adaptive dynamic programming, which are computationally intensive and difficult to implement in practical applications. Simultaneously, within the existing all-drive system framework, fault-tolerant control strategies primarily serve to ensure system stability, failing to simultaneously achieve the dual objectives of system optimization and fault-tolerant control in situations such as actuator failures. Therefore, there is an urgent need to design a fault-tolerant game mechanism and control strategy solution algorithm that coordinates the inner and outer loops for quadrotor helicopter swarm systems, reducing the computational burden of control strategies while improving the intelligence and safety of the quadrotor helicopter swarm system. Summary of the Invention

[0004] Purpose of the invention: To address the above shortcomings, this invention provides a fault-tolerant game-theoretic control method for quadcopter helicopter swarm formations, which achieves synchronous optimization of the inner and outer loops of attitude tracking and formation control, ensuring that the swarm system can maintain stability and optimal control performance under fault conditions, and ultimately achieving interactive Steinberg equilibrium of the system under the cooperation of inner and outer loops.

[0005] Technical Solution: To solve the above problems, this invention discloses a fault-tolerant game-theoretic control method for quadcopter helicopter swarm formations, specifically including the following steps:

[0006] (1) Design the formation mode of the quadrotor helicopter cluster, construct the information interaction topology diagram of the quadrotor helicopter cluster, and establish the fault model of the rotation subsystem and translation subsystem of a single quadrotor helicopter in the quadrotor helicopter cluster.

[0007] (2) Based on the single quadcopter fault model, an adaptive fault diagnosis observer for the quadcopter rotation subsystem is designed, and an adaptive fault estimator is designed based on the reconstructed translation subsystem.

[0008] (3) Taking the translational subsystem of the quadcopter as the leader and the rotational subsystem as the follower, and combining the quadcopter cluster formation design, we construct an inner and outer loop game model based on incentive-based Steinberg differential graph game.

[0009] (4) Using the all-drive system approach, a fault-tolerant game controller is introduced. The controllers for the rotation subsystem and the translation subsystem are designed separately, and the linear closed-loop rotation subsystem and the linear closed-loop translation subsystem of the quadcopter are constructed.

[0010] (5) Design the performance index of the linear closed-loop rotating subsystem of the quadcopter under actuator failure; based on the fault estimate, calculate the optimal fault-tolerant game control strategy of the linear closed-loop rotating subsystem of the quadcopter to realize the attitude tracking control of the quadcopter in the cluster under actuator failure.

[0011] (6) Design performance indicators that can meet the requirements of quadrotor helicopters to achieve formation control under actuator failure. At the same time, based on the fault estimate obtained in step (2), calculate the optimal fault-tolerant game control strategy of quadrotor helicopter translation subsystem to achieve cluster formation control of quadrotor helicopters under actuator failure.

[0012] Furthermore, the implementation process of step (1) is as follows:

[0013] (1.1) Construct a quadcopter cluster with n quadcopter helicopters. In the cluster, one quadcopter helicopter is set as the leader and the rest of the quadcopter helicopters are set as followers. The followers follow the leader to form a formation.

[0014] (1.2) Graph theory is used to describe the information interaction of each quadrotor helicopter in the cluster, and the information interaction topology graph of the entire quadrotor helicopter cluster is constructed; wherein, the information interaction topology graph within the cluster is defined as: Vertex set Represents a node and edge set for a quadcopter helicopter. The diagram illustrates the information exchange between quadcopter helicopters. It is connected; in the vertex set In the list, the leader quadcopter is marked with 0, and the remaining labels are assigned to the follower quadcopters.

[0015] (1.3) The rotation and translation subsystems of the quadcopter i are established as follows:

[0016]

[0017] In the formula, v i For the control input of the quadcopter i; u i The lift of the quadcopter i; For the Euler angle of the quadcopter i, φ i ,θ i ,ψ i These are the roll angle, pitch angle, and yaw angle of the quadcopter i, respectively. Let be the moment of inertia matrix of quadcopter i. These are the moments of inertia of the quadcopter i in the x, y, and z axes, respectively. This indicates the position of quadcopter i. These represent the positions of quadcopter i along the x, y, and z axes, respectively. g is the acceleration due to gravity; The control input matrix for the quadcopter i is the allocation matrix for the four rotors. Let be the parameter matrix of the quadcopter i, which is represented as follows:

[0018]

[0019] In the formula, γ i δ is the distance between the rotor and the center of mass of the quadcopter i. i ε is the drag coefficient of quadcopter i; i m is the propeller correlation coefficient for quadcopter i; i The mass of the quadcopter i;

[0020] (1.4) Construct the rotation and translation subsystems of quadcopter i under actuator failure conditions:

[0021]

[0022] In the formula, The impact of actuator failure on the input of the rotary subsystem of a quadcopter i. The effect of actuator failure on the translation subsystem input of quadcopter i is given, where:

[0023]

[0024] In the formula, I4 is a 4×4 identity matrix. ω is the square of the rotational speed of the quadcopter i. i,1 ,ω i,2 ,ω i,3 ,ω i,4This indicates the rotational speed of the four rotors of the quadcopter i; Let σ be a diagonal matrix consisting of fault values. i,1 ,σ i,2 ,σ i,3 ,σ i,4 These represent the degree of unknown failure of the four rotors of the quadcopter i; Specifically, it is expressed as follows:

[0025]

[0026] In the formula,

[0027] Furthermore, the implementation process of the adaptive observer for the quadcopter helicopter rotation subsystem described in step (2) is as follows:

[0028] Reconstructing the i-rotation subsystem of the quadcopter helicopter yields a system expression that facilitates the application of the adaptive observer method:

[0029]

[0030] In the formula, ρ i The quantity related to actuator failure is denoted as x i1 With x i2 For the state variables of the design, it is represented as L i0 (v i )and The function mapping for the design is represented as I3 is a 3×3 identity matrix; 0 3×3 It is a zero matrix of size 3×3;

[0031] Design an adaptive fault diagnosis observer for the reconfigured quadcopter helicopter rotary subsystem:

[0032]

[0033] In the formula, K i0 With K i1 The feedback gain of the observer; and x i1 x i2 With ρ i The estimated value; 0 3×4 It is a zero matrix of size 3×4.

[0034] Furthermore, the implementation process of designing the adaptive fault estimator in step (2) is as follows:

[0035] Construct the dynamic equations for the state estimation bias:

[0036]

[0037] In the formula, The state estimation bias is expressed as The fault estimation bias is expressed as K i With M i Let be the gain matrix, represented as

[0038] The adaptive fault estimator for quadrotor helicopter i is designed as follows:

[0039]

[0040] In the formula, Π i Let be the weight matrix, satisfying α i Let be a positive integer, satisfying S i For a matrix to satisfy a specific condition, the condition is: S i >0 and Δ i For any number greater than 0;

[0041] Using the obtained adaptive fault estimator, the fault impact values ​​of the rotation and translation subsystem are constructed. and The estimated values ​​are as follows:

[0042]

[0043] The convergence of the quadcopter fault estimator is guaranteed by the following assumptions: there are two unknown constants. and For ρ i The upper limit of absolute value, For ρ i The upper limit of the absolute value of the derivative, the estimation error satisfies the condition. as well as

[0044] Under the assumed conditions, the fault estimator can guarantee and Converging to set gather Defined as:

[0045]

[0046] In the formula, β i θ i The definition is as follows:

[0047]

[0048] In the formula, λ min (Δ i ) represents the matrix Δ within the parentheses. i The smallest eigenvalue; λ max (S i ) represents the matrix S within the parentheses. i The largest eigenvalue, The matrix in parentheses The maximum eigenvalue; min(b) i1 ,b i2 The value of ) is b i1 b i2 The smaller of the two values; For λ max (S i ), The larger value.

[0049] Furthermore, the implementation process of step (3) is as follows:

[0050] Using the quadcopter's rotation subsystem as the inner loop follower, fault-tolerant attitude tracking control is achieved; using the quadcopter's translation subsystem as the outer loop leader, fault-tolerant formation control is achieved; combining the settings of the leader helicopter and follower helicopters in the quadcopter formation and the mutual cooperation relationship of the helicopters in the formation, an inner and outer loop game framework based on incentive-based Steinberg differential graph game is constructed.

[0051] Furthermore, the implementation process of step (4) is as follows:

[0052] (4.1) Design a controller for the rotary subsystem of the quadcopter i:

[0053]

[0054] In the formula, ζ i For fault-tolerant game controllers of rotating subsystems;

[0055] controller v i Substituting the rotating subsystem model, we construct a linear closed-loop rotating subsystem:

[0056]

[0057] The linear closed-loop rotating subsystem is reconstructed, and the reconstructed system expression is as follows:

[0058]

[0059] In the formula, ∈ represents the reconstructed state variable, expressed as: The parameter matrix for the design is represented as follows 0 6i×3 It is a zero matrix of size 6i×3, 0 6(n-i)×3 It is a zero matrix of size 6 (ni) × 3; for The vector formed by the 0th and 1st derivatives is denoted as: ∈( 0～1) Let be the vector formed by the 0th and 1st derivatives of ∈, denoted as .

[0060] (4.2) Design a controller for the translation subsystem of the quadcopter i:

[0061]

[0062] In the formula, η i For a fault-tolerant game controller of a rotating subsystem, it has the following form:

[0063]

[0064] In the formula, p is the state variable of the design, expressed as p (0～1) Let p be the vector formed by the 0th and 1st derivatives of p, denoted as... and Let u be a parameter matrix, which makes the policy u i Graph play is an acceptable strategy for the leader player in a formation;

[0065] controller u i Substituting the translation subsystem model, construct a linear closed-loop translation subsystem:

[0066]

[0067] Reconstructing the linear closed-loop translation subsystem yields the following system expression:

[0068]

[0069] In the formula, The parameter matrix for the design is represented as follows for The vector formed by the 0th and 1st derivatives is denoted as:

[0070] Furthermore, the implementation process of step (5) is as follows:

[0071] (5.1) The performance indicators of the inner ring rotating subsystem of the quadcopter i are designed as follows:

[0072]

[0073] In the formula, ∈ r The desired tracking attitude angle is denoted as... ∈ ir η is the desired tracking attitude angle for the quadcopter i. i For the fault-tolerant game-theoretic controller of the i-ring translation subsystem of a quadcopter; Q i ,R i , Indicates performance index J i The weight matrix in, and Q. i ≥0,R i >0,

[0074] (5.2) The fault-tolerant game-theoretic control strategy model for the inner loop rotating subsystem of quadcopter i is designed as follows:

[0075]

[0076] The constraints are:

[0077]

[0078] (5.3) The optimal fault-tolerant game control strategy for the inner loop rotating subsystem of quadcopter i is calculated and obtained as follows:

[0079]

[0080] Where, λ i For costate variables, λ i satisfy:

[0081]

[0082] In the formula, For the design parameter matrix,

[0083] (5.4) Transforming the optimal fault-tolerant game control strategy of the inner loop rotating subsystem of quadcopter i into a state feedback form, we obtain:

[0084]

[0085] In the formula, P iThis is the solution to the asymmetric Riccati differential equation for the inner ring rotating subsystem of a quadcopter i. The solution to the adjoint equation of the inner ring rotating subsystem of the quadcopter i is given, where the asymmetric Riccati equation and the adjoint equation are as follows:

[0086]

[0087] In the formula, The parameter matrix for the design of a quadcopter helicopter l is represented as follows:

[0088] Furthermore, the implementation process of step (6) is as follows:

[0089] (6.1) The performance indicators of the outer ring translation subsystem of the quadrotor helicopter i are designed as follows:

[0090]

[0091] In the formula, p * The vector consisting of the target positions of all members of a quadcopter formation is denoted as: G represents the desired tracking attitude angle for the quadcopter i. i D i , Indicates performance index Γ i The weight matrix in the matrix, and G. i >0,D i >0; in the matrix In the cluster formation, for the leader, i = 0, W0 is the weight matrix, W0≥0, and for followers in a cluster formation, i≠0.

[0092]

[0093] In the formula, [W i ] nm (n∈(1,n+1),m∈(1,n+1)) is the weight matrix set;

[0094] (6.2) The fault-tolerant game control strategy model of the translation subsystem of quadcopter i is as follows:

[0095]

[0096] The constraints are:

[0097]

[0098] (6.3) The optimal fault-tolerant game control strategy for the outer ring translation subsystem of quadcopter i is calculated and the formula is:

[0099]

[0100] in, Let them be costate variables, which satisfy:

[0101]

[0102] In the formula, Let be a parameter matrix, represented as

[0103] (6.4) Transforming the optimal fault-tolerant game control strategy of the outer ring translation subsystem of quadcopter i into a state feedback form, we obtain:

[0104]

[0105] In the formula, ξ i To and The relevant Lagrange multipliers; Y i With [Y] i For the design parameter matrix, For the solution of the asymmetric Riccati differential equation, The solution to the adjoint equation is given by the following: The asymmetric Riccati equation and the adjoint equation are as follows:

[0106]

[0107] In the formula, Failure impact estimates for all quadcopter helicopters and The vector formed The desired position p for a quadcopter * With expected Euler angles ∈ r The vector formed is represented as Φ, F, It satisfies the following differential equation:

[0108]

[0109] In the formula, f represents the failure impact value of all quadcopter helicopters. and The vector formed is represented as Ξ is the result of passing through the costate variable λi , With Lagrange multipliers β i Defined state variables, Where β i To and The relevant Lagrange multipliers; treating the differential equation as a coupled system where two state variables are each other's inputs. State variables The state matrix, where Φ is the state variable. The control input matrix, where F is the state variable. For the fault distribution matrix of vector f, Let Ξ be the state matrix of the state variable. The control input matrix is ​​the state variable Ξ. For a system with state variable Ξ, for a vector The external input matrix.

[0110] Beneficial Effects: Compared with existing technologies, the present invention offers the following advantages: First, by constructing an inner and outer loop game framework based on an incentive-based Steinberg differential graph game, the present invention reveals the bidirectional interaction mechanism between inner loop attitude tracking and outer loop formation control. Second, the designed adaptive fault observer enables real-time estimation of actuator faults, allowing dynamic adjustment of the control strategy to ensure system stability when a fault occurs. Finally, based on the fault estimation results of the adaptive observer, a fault-tolerant game control strategy for both inner and outer loops is designed. Simultaneously, based on the theory of all-drive systems, the fault-tolerant game strategy is represented in the form of state feedback, transforming the solution of the Hamilton-Jacob equation into the solution of the asymmetric Riccati differential equation, significantly reducing computational complexity and eliminating dependence on learning algorithms. The present invention overcomes the limitations of existing technologies that only consider the unidirectional influence of inner and outer loops, effectively achieving coordinated control of inner and outer loops, enhancing the system's fault tolerance, and optimizing the synchronous performance of inner and outer loops. The game strategy designed in this invention can not only achieve stable attitude tracking and formation control under fault conditions but also realize interactive Steinberg equilibrium. Attached Figure Description

[0111] Figure 1 This is a flowchart of the present invention;

[0112] Figure 2 This is a schematic diagram of a quadcopter helicopter model;

[0113] Figure 3 A graph showing the variation of fault estimation error under the incentive-based Steinberg differential graph game method;

[0114] Figure 4 The graph shows the change in state estimation error under the incentive-based Steinberg differential graph game method.

[0115] Figure 5 A formation trajectory diagram of quadcopter helicopters under the incentive-based Steinberg differential graph game method;

[0116] Figure 6 The attitude angle variation diagram of a quadcopter helicopter swarm under the incentive-based Steinberg differential graph game method;

[0117] Figure 7 The diagram shows the control input variation of the inner loop rotating subsystem of a quadrotor helicopter swarm under the incentive-based Steinberg differential graph game method.

[0118] Figure 8 The diagram shows the control input variation of the outer loop translation subsystem of a quadrotor helicopter swarm under the incentive-based Steinberg differential graph game method.

[0119] Figure 9 A diagram showing the positional changes of a quadcopter helicopter swarm under an incentive-based Steinberg differential graph game method.

[0120] Figure 10 The figure shows the attitude tracking error variation during the formation process of a quadcopter helicopter swarm under the incentive-based Steinberg differential graph game method. Detailed Implementation

[0121] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0122] like Figure 1 As shown, this invention provides a fault-tolerant game-theoretic control method for quadcopter helicopter swarm formations, specifically including the following steps:

[0123] Step 1: Design the formation mode of the quadrotor helicopter cluster, construct the information interaction topology diagram of the quadrotor helicopter cluster, and establish the fault model of the rotation subsystem and translation subsystem of the individual quadrotor helicopter in the quadrotor helicopter cluster.

[0124] Construct a quadcopter cluster with n quadcopters. In the cluster, one quadcopter is designated as the leader and the rest are designated as followers, thus forming a formation with the leader.

[0125] Graph theory is used to describe the information interactions between each quadrotor helicopter in the cluster, thus obtaining the information interaction topology graph of the entire quadrotor helicopter cluster; wherein, the information interaction topology graph within the cluster is defined as: Vertex set Represents a node and edge set for a quadcopter helicopter. This represents the information exchange between quadcopter helicopters; in the vertex set In the graph, the leader quadcopter is labeled 0, and the remaining labels {1,…,n} are assigned to the follower nodes, and it is assumed that the graph… It's connected.

[0126] Considering the dual-time-scale characteristics of the inner and outer rings of a quadrotor helicopter, the rotation and translation subsystems of the quadrotor helicopter are established separately. The rotating subsystem is:

[0127]

[0128] In the formula, For the Euler angle of the quadcopter i, φ i ,θ i ,ψ i These are the roll angle, pitch angle, and yaw angle of the quadcopter i, respectively. Let be the moment of inertia matrix of quadcopter i; The input torque for quadcopter i is expressed as:

[0129]

[0130] In the formula, γ i δ is the distance between the rotor and the center of mass of the quadcopter i. i ε is the drag coefficient of quadcopter i; i For the propeller-related proportionality factor of quadcopter i; The control inputs for the roll angle, pitch angle, and yaw angle of the quadcopter i are respectively expressed as:

[0131]

[0132] In the formula, ω i,1 ,ω i,2 ,ω i,3 ,ω i,4 This indicates the rotational speed of the four rotors of a quadcopter.

[0133] The control inputs for Euler angles are distributed to the four rotors via a distribution board. By using a distribution matrix, the quadcopter helicopter's rotary subsystem can be transformed into the following form:

[0134]

[0135] In the formula, the allocation matrix of quadcopter i is...

[0136] By transforming the rotation subsystem of the quadcopter, the control variable settings are simplified, thereby achieving the control objective more efficiently. Simultaneously, to avoid singular value phenomena in the Euler angle expression, the Euler angle variation of the quadcopter needs to be restricted. The restriction condition is: |φ i |<π / 2,|θ i |<π / 2,|ψi |<π / 2.

[0137] quadcopter helicopter The translation subsystem is as follows:

[0138]

[0139] In the formula, These are the x, y, and z coordinates of the quadcopter i in the geodetic coordinate system. They are respectively The second derivative; m i Let i be the mass of the quadcopter i; g is the acceleration due to gravity. The lift obtained by the quadcopter i is expressed as:

[0140]

[0141] By redefining the control input, the translation subsystem of a quadcopter helicopter can be represented as follows:

[0142]

[0143] In the formula, The parameter matrix for the design of quadcopter i is represented as follows: This indicates the position of quadcopter i. Where g is the acceleration due to gravity; u i The redefined control input is represented as follows:

[0144]

[0145] By transforming the translation subsystem of a quadcopter, the control settings are simplified, thereby achieving the control objectives more efficiently.

[0146] Considering the possibility of actuator failure during quadcopter flight:

[0147]

[0148] In the formula, σ represents the square of the rotor speed of the quadcopter UAV i under fault conditions. i,s To satisfy the condition 0 < σ i,s Unknown fault severity ≤ 1; assuming condition σ is met. i,s =1, then the quadcopter i is healthy; if the condition 0 < σ is satisfied. i,s If the value is less than 1, then the quadcopter i has experienced an actuator failure.

[0149] Considering the impact of actuator failure on the rotation and translation subsystems of a quadcopter helicopter, the subsystem models can be represented as follows:

[0150]

[0151] In the formula, The impact of actuator failure on the input of the rotary subsystem of a quadcopter i. The effect of an actuator failure in quadcopter i on the input of the translation subsystem is expressed as:

[0152]

[0153] In the formula, I4 is a 4×4 identity matrix. Let ω be the square of the rotational speed of the quadcopter i, where ω i,1 ,ω i,2 ,ω i,3 ,ω i,4 The rotational speeds of the four rotors of a quadcopter i, the positions of the four rotors, and their directions of rotation are as follows: Figure 2 As shown; Let σ be a diagonal matrix consisting of fault values. i,1 ,σ i,2 ,σ i,3 ,σ i,4 These represent the degree of unknown failure of the four rotors of the quadcopter UAV i. Specifically, it is expressed as follows:

[0154]

[0155] In the formula,

[0156] By introducing a fault model of a quadcopter, we can study the control methods of quadcopter under fault conditions.

[0157] Step 2: Design an adaptive observer for the quadcopter and estimate unknown faults based on the obtained quadcopter fault model.

[0158] Design an adaptive observer for the rotational subsystem of a quadcopter; reconstruct the rotational subsystem of the quadcopter to obtain a system expression that facilitates the application of the adaptive observer method.

[0159] By defining matrix L i0 (v i )and This can be simplified to obtain a new system expression:

[0160]

[0161] In the formula, the defined matrix is:

[0162] Next, redefine the system state variables. as well as The system was refactored as follows:

[0163]

[0164] In the formula, ρ i The quantity related to actuator failure is expressed as: I3 is a 3×3 identity matrix; 0 3×3 It is a zero matrix of size 3×3.

[0165] Design an adaptive fault diagnosis observer for the reconfigured quadcopter helicopter rotary subsystem:

[0166]

[0167] Among them, K i0 With K i1 The feedback gain of the observer; and x i1 x i2 With ρ i The estimated value.

[0168] An adaptive fault estimator is designed based on an adaptive observer and a reconstructed translational subsystem; a dynamic equation concerning the state estimation bias is constructed:

[0169]

[0170] In the formula, The state estimation bias is expressed as The fault estimation bias is expressed as K i With M i The gain matrix is ​​represented as follows:

[0171] The following is a design for a fault estimator for quadcopter helicopters:

[0172]

[0173] In the formula, Π i Let be the weight matrix, satisfying α i Let be a positive integer, satisfying Si For a matrix to satisfy a specific condition, the condition is: S i >0 and Δ i Let be any number greater than 0.

[0174] Using the obtained adaptive fault estimator, the fault impact values ​​of the rotation and translation subsystem are constructed. and The estimated values ​​are as follows:

[0175]

[0176] It is important to note that, in order to ensure the convergence of the quadcopter fault estimator, the following assumptions need to be made: there are two unknown constants. and For ρ i The upper limit of absolute value, For ρ i The upper limit of the absolute value of the derivative, the estimation error satisfies the condition. as well as

[0177] Under the assumed conditions, the fault estimator can guarantee and Converging to set gather Defined as:

[0178]

[0179] In the formula, β i θ i The definition is as follows:

[0180]

[0181] In the formula, λ min (Δ i ) represents the matrix Δ within the parentheses. i The smallest eigenvalue; λ max (S i ) represents the matrix S within the parentheses. i The largest eigenvalue, The matrix in parentheses The maximum eigenvalue; min(b) i1 ,b i2 The value of ) is b i1 b i2 The smaller of the two values; For λ max (S i ), The larger value.

[0182] Under the action of the adaptive observer, it can be guaranteed that the state estimation bias and the fault estimation bias converge to a specific set. Ensure the effectiveness of the fault-tolerant game controller introduced in step four.

[0183] Step 3: In the quadrotor helicopter system, the quadrotor helicopter translation subsystem is taken as the leader and the rotation subsystem is taken as the follower. Combined with the quadrotor helicopter swarm formation design, an inner and outer loop game framework based on incentive-based Steinberg differential graph game is constructed.

[0184] In the constructed incentive-based Steinberg differential graph game framework, the translational subsystem of the quadcopter helicopter acts as the leader player, and the rotational subsystem acts as the follower player. Specifically, the outer-loop translational subsystem announces its control strategy u to the inner-loop rotational subsystem in advance. i This helps the rotating subsystem calculate the required thrust. and the desired attitude angle θ ir φ ir In order to achieve attitude tracking control.

[0185] Follower players follow the control strategy provided by the leader player. i The system aims to make the optimal response, minimize attitude tracking errors, and achieve Nash equilibrium in the graph game of the follower players. Simultaneously, by leveraging the fast timescale characteristics of the inner loop to establish an adaptive fault estimator, the controllers for both the inner and outer loops can be reconfigured in a timely manner, thereby improving the system's fault tolerance.

[0186] Based on the best response of the follower players and combined with the quadcopter helicopter swarm formation design, the leader player designs the optimal game strategy to achieve fault-tolerant formation control and ensure the achievement of interactive Steinberg equilibrium within the incentive-based Steinberg differential graph game framework.

[0187] Using game theory, the cooperative relationship between quadcopters is formalized into a graph game to achieve formation; proactive fault-tolerant control is achieved through Steinberg game theory. Furthermore, the interaction between the outer-loop leader and inner-loop followers in the quadcopter system is effectively optimized through the constructed incentive-based Steinberg differential graph game mechanism. It is important to note that the leader player and the leader quadcopter are two distinct concepts in the design. The leader player refers to the role of the player in the incentive-based Steinberg differential graph game, interacting with the follower players to form a complete game structure. In this case, the translational subsystems of all quadcopters act as leaders in the incentive-based Steinberg differential graph game; the leader quadcopter is a physical component of the formation, responsible for guiding the following quadcopters to the desired destination.

[0188] Step 4: Using the all-drive system approach, introduce a fault-tolerant game controller and design controllers for the rotation subsystem and translation subsystem of the quadcopter respectively, to obtain the linear closed-loop rotation subsystem and linear closed-loop translation subsystem of the quadcopter.

[0189] Considering the nonlinear characteristics of the quadrotor helicopter subsystem model, the all-drive system method is used to handle the rotation and translation subsystems of the quadrotor helicopter separately.

[0190] Design a controller for a rotating subsystem. i :

[0191]

[0192] In the formula, ζ i It is a fault-tolerant game controller for rotating subsystems.

[0193] controller v i Substituting the rotating subsystem model, we construct a linear closed-loop rotating subsystem:

[0194]

[0195] The linear closed-loop rotating subsystem is reconstructed, and the reconstructed system expression is as follows:

[0196]

[0197] In the formula, ∈ represents the reconstructed state variable, expressed as: The parameter matrix for the design is represented as follows 0 6i×3 It is a zero matrix of size 6i×3, 0 6(n-i)×3 It is a zero matrix of size 6 (ni) × 3; for The vector formed by the 0th and 1st derivatives is denoted as: ∈ (0～1) Let be the vector formed by the 0th and 1st derivatives of ∈, denoted as .

[0198] By introducing a linear closed-loop rotating subsystem, the subsequent design of performance indicators and the solution of control variables become simpler and more efficient.

[0199] Design a controller for the translation subsystem. i :

[0200]

[0201] In the formula, η i It is a fault-tolerant game controller for the translation subsystem.

[0202] controller u i Substituting into the translation subsystem model, we can obtain the linear closed-loop expression for the translation subsystem:

[0203]

[0204] Reconstructing the linear closed-loop translation subsystem yields the following system expression:

[0205]

[0206] In the formula, The parameter matrix for the design is represented as follows p is the reconstructed state variable, represented as p (0～1) Let p be the vector formed by the 0th and 1st derivatives of p, denoted as... for The vector formed by the 0th and 1st derivatives is denoted as:

[0207] By introducing a linear closed-loop translation subsystem, the subsequent design of performance indicators and the solution of control variables become simpler and more efficient.

[0208] Step 5: Design the performance index of the quadcopter rotation subsystem in the cluster under the fault-tolerant game problem, and calculate the optimal fault-tolerant game control strategy of the quadcopter rotation subsystem based on the fault estimate.

[0209] Define the control strategy for transmission in the outer loop translation subsystem. i The form is:

[0210]

[0211] In the formula, u xi ,u yi ,u zi These are the control inputs for the translation subsystem of a quadcopter in the x-axis, y-axis, and z-axis directions, respectively. and Let u be a parameter matrix, which makes the policy u i Graph play is an acceptable strategy for the leader player in a formation.

[0212] Based on control strategy u i The thrust required by the inner rotating subsystem was calculated. And the desired pitch angle θ ir and roll angle φ ir :

[0213]

[0214] Based on the inner and outer loop game framework of incentive-based Steinberg differential graph game, the performance index of the i-th quadcopter helicopter inner loop rotation subsystem is designed as follows:

[0215]

[0216] In the formula, ∈ r The desired tracking attitude angle is denoted as... ∈ ir Q represents the desired tracking attitude angle for the quadcopter i; i ,R i , Indicates performance index J i The weight matrix in, and Q. i ≥0,R i >0,

[0217] Design a game-theoretic control strategy model for an inner-loop rotating subsystem. Specifically, the attitude tracking problem of the inner-loop rotating subsystem under actuator failure can be described as: given η i ,

[0218]

[0219] The constraints are:

[0220]

[0221] Based on the estimated fault obtained from the fault estimator, the optimal fault-tolerant game control strategy for the inner ring rotating subsystem of a quadcopter is calculated.

[0222] Define the Hamiltonian function of the rotating subsystem of helicopter i as:

[0223]

[0224] Where, λ i Let be the costate variables of the i-th quadcopter helicopter rotation subsystem; optimal fault-tolerant game control strategy. satisfy:

[0225]

[0226] Using the principle of minimum values, we can obtain:

[0227]

[0228] Therefore, the optimal fault-tolerant game-theoretic control strategy for the i-th quadcopter rotor subsystem can be obtained. for:

[0229]

[0230] Where, λ i satisfy:

[0231]

[0232] In the formula,

[0233] Inner-loop optimal fault-tolerant game control strategy Convert to a state feedback expression. Set the costate variable λ. i It is in the following form:

[0234]

[0235] In the formula, P i This is a solution to the asymmetric Riccati differential equation; The solution to the adjoint equation is given; the asymmetric Riccati differential equation and its adjoint equation are as follows:

[0236]

[0237] In the formula,

[0238] The state feedback form of the fault-tolerant game control strategy for the inner-loop rotating subsystem is obtained as follows:

[0239]

[0240] To transform the fault-tolerant game strategy into a state feedback form, an adjoint equation is introduced, and then fault estimation is used in the adjoint equation. Replace unknown fault Thus through Fault compensation was implemented, enhancing the system's fault tolerance and enabling attitude tracking control under fault conditions in the inner loop. Simultaneously, the feedback matrix P... i With fault compensation signal All of these are coupled with the design of the adjacent quadcopter helicopters, reflecting the coordinated control among the quadcopter helicopters in the cluster, which effectively improves the overall stability and fault tolerance of the system under actuator failure conditions.

[0241] Step 6: For quadcopter swarms, design the performance indicators of the translation subsystems of the leader and follower quadcopters in the swarm formation under the fault-tolerant game problem and swarm formation task. At the same time, integrate the performance indicators of the translation subsystems of the leader and follower quadcopters in the swarm formation into a single performance indicator. Based on the fault estimate, calculate the optimal fault-tolerant game control strategy for the translation subsystems of the quadcopters in the swarm.

[0242] Considering that the leader and followers in a quadcopter swarm have different objectives, performance indicators are designed separately for the leader and follower quadcopters. The leader quadcopter needs to optimize its trajectory to reach its desired destination; therefore, its performance indicators are as follows:

[0243]

[0244] In the formula, W0, G0, and D0 are the weight matrices of the performance index Γ0, and W0≥0, G0>0, and D0>0.

[0245] Meanwhile, the follower quadcopter i is responsible for maintaining the formation shape, therefore its performance specifications are:

[0246]

[0247] In the formula, χ i0 χ represents the relative positions of the i-th follower quadcopter and the leader quadcopter, respectively; il Let W be the relative position of the i-th follower quadcopter and the j-th follower quadcopter; i0 W il G i D i Let W be the performance index weight matrix for the i-th follower quadcopter helicopter, and let W be... i0 ≥0, W il ≥0, G i >0, D i >0.

[0248] To ensure consistency in the derivation of control strategies for quadcopters in the cluster, the performance indicators of the leader and follower quadcopters in the formation are unified and integrated by redefining the system state variables and weight matrices.

[0249] Based on the desired destination And formation mode χ ij The numerical relationship transforms the target position of the follower quadcopter helicopter into the following form:

[0250]

[0251] Therefore, the performance specifications of the follower quadcopter helicopter can be written as:

[0252]

[0253] By resetting the system state variables Integrating the performance metrics of the leader and follower quadrotor helicopters into a unified format, the performance metrics of the outer ring translation subsystem of quadrotor helicopter i are designed as follows:

[0254]

[0255] In the formula, G i D i , Indicates performance index Γ i In the weight matrix, for the leader i=0, we have For followers i≠0, we have:

[0256]

[0257] in, satisfy Its elements are defined as:

[0258]

[0259] Design a game-theoretic control strategy model for the outer loop translation subsystem of a quadcopter i. Specifically, the fault-tolerant game problem of the outer loop translation subsystem in a swarm formation task can be described as: given ζ i ,ζ -i ,

[0260]

[0261] The constraints are:

[0262]

[0263] Within the framework of incentive-based Steinberg differential graph games, by introducing two constraints on the inner loop rotation subsystem, the leader's fault-tolerant game control strategy η is made possible. i The solution process incorporates the optimal response of the inner-loop rotating subsystem. and This enables a closed-loop design for both the inner and outer loops, which helps improve the ability of the outer loop strategy to correct attitude tracking errors.

[0264] The optimal fault-tolerant game-theoretic control strategy for the outer-loop translation subsystem of quadrotor helicopter i is obtained through calculation. The Hamiltonian function of the rotation subsystem of quadrotor helicopter i is defined as:

[0265]

[0266] In the formula, For costate variables; β i With ξ i It is a Lagrange multiplier.

[0267] Optimal Fault-Tolerant Game Control Strategy satisfy:

[0268]

[0269] Using the principle of minimum values, we can obtain:

[0270]

[0271] Therefore, the optimal fault-tolerant game control strategy can be obtained. for:

[0272]

[0273] in β i ξ i satisfy:

[0274]

[0275] To enhance the fault tolerance of quadcopter helicopters in the event of actuator failure, an optimal fault-tolerant game-theoretic control strategy is adopted. It is a form of status feedback.

[0276] First, define the following state variables:

[0277]

[0278] Next, using the defined state variables, the following differential equation is derived:

[0279]

[0280] In the formula, the differential equation is regarded as a coupled system in which two state variables are mutually input, and f is the fault influence value of all quadcopter helicopters. and The vector formed The desired position p for a quadcopter * With expected Euler angles ∈ r The vector formed State variables The state matrix, where Φ is the state variable. The control input matrix, where F is the state variable. For the fault distribution matrix of vector f, Let Ξ be the state matrix of the state variable. The control input matrix is ​​the state variable Ξ. For a system with state variable Ξ, for a vector The external input matrix, as described above, is specifically represented as follows:

[0281]

[0282]

[0283] In the formula,

[0284] At the same time, matrix It has the following affine relation with matrix Ξ:

[0285]

[0286] In the formula, For the solution to the corresponding asymmetric Riccati equation:

[0287]

[0288] Satisfying the corresponding adjoint equation:

[0289]

[0290] In the formula, For the estimate of f, is the estimated failure impact value for all quadcopter helicopters. and The vector formed, and has

[0291] Therefore, the fault-tolerant game controller of the outer loop translation subsystem of the quadcopter i can be represented by the following state feedback form:

[0292]

[0293] In the formula,

[0294] The fault-tolerant game strategy is transformed into a state feedback form, and an adjoint equation is introduced. Then, the estimated fault is used in the adjoint equation. Replace the unknown fault f, thereby passing Achieve fault compensation.

[0295] The present invention can calculate the fault-tolerant game controller of the inner ring rotation subsystem and the outer ring translation subsystem of all quadrotor helicopters in the cluster, thereby realizing attitude tracking and formation control of all quadrotor helicopters in the event of a fault during the process of the formation cluster reaching the predetermined position.

[0296] This implementation uses a cluster of three quadcopter helicopters to verify the effectiveness of the proposed fault-tolerant game control method, i.e., the cluster point set.

[0297] First, the physical parameters of the quadcopter, including initial velocity, initial angle, desired position, and formation mode, are set. A simulation experiment of the quadcopter is then conducted based on the following physical parameters:

[0298] m = 2kg

[0299] g = 9.81 m / s 2

[0300]

[0301] The initial positions of the three quadcopter helicopters are set as follows:

[0302] p0(0)=[0.5485,0.1653,0.4939] T

[0303] p1(0)=[0.0263,0.3188,0.5330] T

[0304] p2(0)=[0.1349,0.9138,0.6406] T

[0305] Set the initial angle of the three quadcopter helicopters to 0:

[0306]

[0307] The desired destination for the leader of the quadcopter formation is:

[0308]

[0309] The formation mode for a quadcopter helicopter swarm is set as follows:

[0310] χ 10 =χ 21 =[1,1,3] T

[0311] χ 20 =[2,2,6] T

[0312] Next, determine the timing and severity of the quadcopter's actuator failure. Assume the quadcopter experiences an actuator failure at the 10th second, with the following severity:

[0313] Λ0=0.5I4

[0314] Λ1=0.6I4

[0315] Λ2=0.7I4

[0316] Then, set the weight matrix in the performance metrics. For the i-th drone, The following weight matrix design is proposed:

[0317] Q1 = diag(513,0) 3×3 ,0 3×3 )

[0318] Q2 = diag(0 3×3 ,5I 3×3 ,0 3×3 )

[0319] Q3 = diag(0 3×3 0 3×3 ,5I 3×3 )

[0320]

[0321] Based on the parameters set above, the simulation experiment will begin.

[0322] To demonstrate the correlation between the changes in simulation results and fault-tolerant game strategies, the simulation process was divided into two stages based on the time of the fault occurrence: the first stage was the simulation of the quadcopter cluster before the fault occurred; the second stage was the simulation of the quadcopter cluster after the fault occurred.

[0323] The first phase of the simulation experiment is as follows: Before the fault occurs, since there is no actuator fault in the system, the system... and It is 0. Figure 5 The simulation demonstrates the trajectory changes of a cluster of quadcopters. Figure 9 The simulation demonstrates the coordinate changes of all helicopters in a quadcopter swarm. The simulation results show that, with the designed controller, the quadcopters achieved the desired formation before the malfunction occurred, and each quadcopter reached its designated position. Figure 9 The target location is shown in the figure.

[0324] The second phase of the simulation experiment is as follows: the fault occurs at 10 seconds, from... Figure 5 It can be seen that the quadcopter formation deviated to varying degrees. Figure 3 The variation of the difference between the fault estimate and the actual fault by the adaptive observer during simulation is shown. Simulation results indicate that the fault estimation bias gradually converges to the neighborhood near zero, demonstrating the effectiveness of the fault estimator. Figure 4 The simulation results show the variation in the deviation between the estimated and actual state variables of the adaptive observer. The simulation results in the figure demonstrate that the state estimation deviation gradually converges to a neighborhood near zero, illustrating the effectiveness of the adaptive observer's state estimation. Figure 6The simulation demonstrates the attitude angle changes of each quadrotor in a quadrotor cluster. The simulation results show that after a fault occurs, the attitude angles of all quadrotors smoothly transition and eventually stabilize, illustrating the effectiveness of the fault-tolerant game controller. Figure 7 and Figure 8 The numerical changes of the fault-tolerant game controllers for the inner and outer loops of the quadrotor helicopter swarm design are shown in the simulation. The simulation results in the figure show that each quadrotor helicopter in the swarm can maintain stable fuselage attitude and hover at the desired position after completing the mission. Figure 9 The simulation demonstrates the positional changes of a quadcopter swarm. The simulation results in the figure show that each quadcopter in the swarm can reach the desired position and achieve the formation objective. Figure 10 The simulation results show the attitude tracking error changes of a quadcopter swarm during formation. The simulation results in the figure show that the attitude deviation of the quadcopters in the swarm continuously decreases and eventually tends to be consistent, demonstrating the effectiveness of the inner-loop attitude tracking control.

[0325] In summary, the method described in this invention can not only ensure that a quadcopter completes its fixed-point flight mission when a malfunction occurs during flight, but also ensure the formation maintenance and attitude stability of a quadcopter swarm.

[0326] The above description provides a detailed account of the method described in this invention, but it is obvious that the specific implementation of this invention is not limited thereto. For those skilled in the art, various obvious modifications made to this invention without departing from the spirit and scope of the claims are within the protection scope of this invention.

Claims

1. A fault-tolerant game-theoretic control method for quadcopter helicopter swarm formation, characterized in that, Includes the following steps: (1) Design the formation mode of the quadrotor helicopter cluster, construct the information interaction topology diagram of the quadrotor helicopter cluster, and establish the fault model of the rotation subsystem and translation subsystem of a single quadrotor helicopter in the quadrotor helicopter cluster. (2) Based on the single quadcopter fault model, an adaptive fault diagnosis observer for the quadcopter rotation subsystem is designed, and an adaptive fault estimator is designed based on the reconstructed translation subsystem. (3) Taking the translational subsystem of the quadcopter as the leader and the rotational subsystem as the follower, and combining the quadcopter cluster formation design, we construct an inner and outer loop game model based on incentive-based Steinberg differential graph game. (4) Using the all-drive system approach, a fault-tolerant game controller is introduced. The controllers for the rotation subsystem and the translation subsystem are designed separately, and the linear closed-loop rotation subsystem and the linear closed-loop translation subsystem of the quadcopter are constructed. (5) Design the performance index of the linear closed-loop rotating subsystem of the quadcopter under actuator failure; based on the fault estimate, calculate the optimal fault-tolerant game control strategy of the linear closed-loop rotating subsystem of the quadcopter to realize the attitude tracking control of the quadcopter in the cluster under actuator failure. (6) Design performance indicators that can meet the requirements of quadcopter formation control under actuator failure. At the same time, based on the fault estimate obtained in step (2), calculate the optimal fault-tolerant game control strategy of quadcopter translation subsystem to realize cluster formation control of quadcopter under actuator failure. The implementation process of step (4) is as follows: (4.1) Design a controller for the rotary subsystem of the quadcopter i: In the formula, ζ i For fault-tolerant game controllers of rotating subsystems; Let be the moment of inertia matrix of quadcopter i. These are the moments of inertia of the quadcopter i in the x, y, and z axes, respectively. controller v i Substituting the rotating subsystem model, we construct a linear closed-loop rotating subsystem: in, The impact of actuator failure on the input of the rotary subsystem of a quadcopter i. The impact of actuator failure on the translation subsystem input of a quadcopter i. The linear closed-loop rotating subsystem is reconstructed, and the reconstructed system expression is as follows: In the formula, ∈ represents the reconstructed state variable, expressed as: The parameter matrix for the design is represented as follows 0 6i×3 It is a zero matrix of size 6i×3, 0 6(n-i)×3 It is a zero matrix of size 6 (ni) × 3; for The vector formed by the 0th and 1st derivatives is denoted as: ∈ (0～1) Let be the vector formed by the 0th and 1st derivatives of ∈, denoted as . (4.2) Design a controller for the translation subsystem of the quadcopter i: In the formula, η i The fault-tolerant game controller for the translation subsystem has the following form: In the formula, p is the state variable of the design, expressed as p (0～1) Let p be the vector formed by the 0th and 1st derivatives of p, denoted as... and Let u be a parameter matrix, which makes the policy u i Graph play is an acceptable strategy for the leader player in a formation; controller u i Substituting the translation subsystem model, construct a linear closed-loop translation subsystem: Reconstructing the linear closed-loop translation subsystem yields the following system expression: In the formula, n represents the number of helicopters in the helicopter cluster. The parameter matrix for the design is represented as follows for The vector formed by the 0th and 1st derivatives is denoted as: The implementation process of step (5) is as follows: (5.1) The performance indicators of the inner ring rotating subsystem of the quadcopter i are designed as follows: In the formula, ∈ r The desired tracking attitude angle is denoted as... ∈ ir Q represents the desired tracking attitude angle for the quadcopter i; i ,R i , Indicates performance index J i The weight matrix in, and has (5.2) The fault-tolerant game-theoretic control strategy model for the inner loop rotating subsystem of quadcopter i is designed as follows: The constraints are: (5.3) The optimal fault-tolerant game control strategy for the inner loop rotating subsystem of quadcopter i is calculated and obtained as follows: Where, λ i For costate variables, λ i satisfy: In the formula, For the design parameter matrix, (5.4) Transforming the optimal fault-tolerant game control strategy of the inner loop rotating subsystem of quadcopter i into a state feedback form, we obtain: In the formula, P i This is the solution to the asymmetric Riccati differential equation for the inner ring rotating subsystem of a quadcopter i. The solution to the adjoint equation of the inner ring rotating subsystem of the quadcopter i is given, where the asymmetric Riccati equation and the adjoint equation are as follows: In the formula, The parameter matrix for the design of a quadcopter helicopter l is represented as follows:

2. The fault-tolerant game-theoretic control method for quadcopter helicopter swarm formation according to claim 1, characterized in that, The implementation process of step (1) is as follows: (1.1) Construct a quadcopter cluster with n quadcopter helicopters. In the cluster, one quadcopter helicopter is set as the leader and the rest of the quadcopter helicopters are set as followers. The followers follow the leader to form a formation. (1.2) Graph theory is used to describe the information interaction of each quadrotor helicopter in the cluster, and the information interaction topology graph of the entire quadrotor helicopter cluster is constructed; wherein, the information interaction topology graph within the cluster is defined as: Vertex set Represents a node and edge set for a quadcopter helicopter. The diagram illustrates the information exchange between quadcopter helicopters. It is connected; in the vertex set In the list, the leader quadcopter is marked with 0, and the remaining labels are assigned to the follower quadcopters. (1.3) The rotation and translation subsystems of the quadcopter i are established as follows: In the formula, v i For the control input of the quadcopter i; u i The lift of the quadcopter i; For the Euler angle of the quadcopter i, φ i ,θ i ,ψ i These are the roll angle, pitch angle, and yaw angle of the quadcopter i, respectively. Let be the moment of inertia matrix of quadcopter i. These are the moments of inertia of the quadcopter i in the x, y, and z axes, respectively. This indicates the position of quadcopter i. These represent the positions of quadcopter i along the x, y, and z axes, respectively. g is the acceleration due to gravity; The control input matrix for the quadcopter i is the allocation matrix for the four rotors. Let be the parameter matrix of the quadcopter i, which is represented as follows: In the formula, γ i δ is the distance between the rotor and the center of mass of the quadcopter i. i ε is the drag coefficient of quadcopter i; i m is the propeller correlation coefficient for quadcopter i; i The mass of the quadcopter i; (1.4) Construct the rotation and translation subsystems of quadcopter i under actuator failure conditions: In the formula, The impact of actuator failure on the input of the rotary subsystem of a quadcopter i. The effect of actuator failure on the translation subsystem input of quadcopter i is given, where: In the formula, I4 is a 4×4 identity matrix. ω is the square of the rotational speed of the quadcopter i. i,1 ,ω i,2 ,ω i,3 ,ω i,4 This indicates the rotational speed of the four rotors of the quadcopter i; Let σ be a diagonal matrix consisting of fault values. i,1 ,σ i,2 ,σ i,3 ,σ i,4 These represent the degree of unknown failure of the four rotors of the quadcopter i; Specifically, it is expressed as follows: In the formula, 3. The fault-tolerant game-theoretic control method for quadcopter helicopter swarm formation according to claim 2, characterized in that, The implementation process of the adaptive observer for the quadcopter helicopter rotation subsystem described in step (2) is as follows: Reconstructing the i-rotation subsystem of the quadcopter helicopter yields a system expression that facilitates the application of the adaptive observer method: In the formula, ρ i The quantity related to actuator failure is denoted as x i1 With x i2 For the state variables of the design, it is represented as L i0 (v i )and The function mapping for the design is represented as I3 is a 3×3 identity matrix; 0 3×3 It is a zero matrix of size 3×3; Design an adaptive fault diagnosis observer for the reconfigured quadcopter helicopter rotary subsystem: In the formula, K i0 With K i1 The feedback gain of the observer; and x i1 x i2 With ρ i The estimated value; 0 3×4 It is a zero matrix of size 3×4.

4. The fault-tolerant game-theoretic control method for quadcopter helicopter swarm formation according to claim 3, characterized in that, The implementation process of designing the adaptive fault estimator in step (2) is as follows: Construct the dynamic equations for the state estimation bias: In the formula, The state estimation bias is expressed as The fault estimation bias is expressed as K i With M i Let be the gain matrix, represented as The adaptive fault estimator for quadrotor helicopter i is designed as follows: In the formula, Π i Let be the weight matrix, satisfying α i Let be a positive integer, satisfying S i For a matrix to satisfy a specific condition, the condition is: S i >0 and Δ i For any number greater than 0; Using the obtained adaptive fault estimator, the fault impact values ​​of the rotation and translation subsystem are constructed. and The estimated values ​​are as follows: The convergence of the quadcopter fault estimator is guaranteed by the following assumptions: there are two unknown constants. and For ρ i The upper limit of absolute value, For ρ i The upper limit of the absolute value of the derivative, the estimation error satisfies the condition. as well as Under the assumed conditions, the fault estimator can guarantee and Converging to set gather Defined as: In the formula, β i θ i The definition is as follows: In the formula, λ min (Δ i ) represents the matrix Δ within the parentheses. i The smallest eigenvalue; λ max (S i ) represents the matrix S within the parentheses. i The largest eigenvalue, The matrix in parentheses The largest eigenvalue; min(b i1 ,b i2 The value of ) is b i1 b i2 The smaller of the two values; For λ max (S i ), The larger value.

5. The fault-tolerant game-theoretic control method for quadcopter helicopter swarm formation according to claim 4, characterized in that, The implementation process of step (3) is as follows: Using the quadcopter's rotation subsystem as the inner loop follower, fault-tolerant attitude tracking control is achieved; using the quadcopter's translation subsystem as the outer loop leader, fault-tolerant formation control is achieved; combining the settings of the leader helicopter and follower helicopters in the quadcopter formation and the mutual cooperation relationship of the helicopters in the formation, an inner and outer loop game framework based on incentive-based Steinberg differential graph game is constructed.

6. The fault-tolerant game-theoretic control method for quadcopter helicopter swarm formation according to claim 5, characterized in that, The implementation process of step (6) is as follows: (6.1) The performance indicators of the outer ring translation subsystem of the quadrotor helicopter i are designed as follows: In the formula, p * The vector consisting of the target positions of all members of a quadcopter formation is denoted as: G represents the desired tracking attitude angle for the quadcopter i. i D i , Indicates performance index Γ i The weight matrix in the matrix, and G. i >0,D i >0; in the matrix In the cluster formation, for the leader, i = 0, W0 is the weight matrix, W0≥0, and for followers in a cluster formation, i≠0. In the formula, [W i ] nm (n∈(1,n+1),m∈(1,n+1)) is the weight matrix set; (6.2) The fault-tolerant game control strategy model of the translation subsystem of quadcopter i is as follows: The constraints are: (6.3) The optimal fault-tolerant game control strategy for the outer ring translation subsystem of quadcopter i is calculated and the formula is: in, Let them be costate variables, which satisfy: In the formula, Let be a parameter matrix, represented as (6.4) Transforming the optimal fault-tolerant game control strategy of the outer ring translation subsystem of quadcopter i into a state feedback form, we obtain: In the formula, ξ i To and The relevant Lagrange multipliers; Y i With [Y] i For the design parameter matrix, For the solution of the asymmetric Riccati differential equation, The solution to the adjoint equation is given by the following: The asymmetric Riccati equation and the adjoint equation are as follows: In the formula, Failure impact estimates for all quadcopter helicopters and The vector formed The desired position p for a quadcopter * With expected Euler angles ∈ r The vector formed is represented as Φ, F, It satisfies the following differential equation: In the formula, f represents the failure impact value of all quadcopter helicopters. and The vector formed is represented as Ξ is the result of passing through the costate variable λ i , With Lagrange multipliers β i Defined state variables, Where β i To and The relevant Lagrange multipliers; treating the differential equation as a coupled system where two state variables are each other's inputs. State variables The state matrix, where Φ is the state variable. The control input matrix, where F is the state variable. For the fault distribution matrix of vector f, Let Ξ be the state matrix of the state variable. The control input matrix is ​​the state variable Ξ. For a system with state variable Ξ, for a vector The external input matrix.

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