Fault-tolerant game control method for four-rotor helicopter cluster formation

Abstract

The invention discloses a fault-tolerant game control method for quad-rotor helicopter cluster formation, and the method comprises the steps: designing a quad-rotor helicopter cluster formation mode, constructing an information interaction topological graph of a helicopter cluster, and building a fault model of a rotation subsystem and a translation subsystem; designing a self-adaptive fault diagnosis observer, and estimating an actuator fault; the translation subsystem is used as a leader, the rotation subsystem is used as a follower, and an inner and outer ring fault-tolerant game framework based on the incentive Steinberg differential graph game is constructed; fault-tolerant game controllers of the rotation subsystem and the translation subsystem are respectively designed through an all-drive system theory, and a linear closed-loop system of the four-rotor helicopter is constructed; and designing performance indexes of the rotation subsystem and the translation subsystem, and designing an optimal fault-tolerant game control strategy based on a differential game theory. According to the invention, stable formation control of the four-rotor helicopter cluster can be realized, the fault-tolerant capability of the helicopter cluster is effectively improved, and the stability and intelligence of the helicopter cluster system are enhanced.

Description

Technical Field

[0001] The invention belongs to the technical field of multi-aircraft control, and in particular relates to a fault-tolerant game control method for a cluster formation of four-rotor helicopters. Background Art

[0002] Quadcopters are widely used in civil and military fields due to their flexible maneuverability and wide range of application scenarios such as forest fire monitoring, rescue missions, logistics distribution, etc. In order to achieve efficient and stable cluster formation control, game theory, as a powerful analysis tool, has been introduced into the control strategy of helicopter clusters. For example: in a multi-agent system, game theory can complete tasks such as optimizing formation control and maintaining formation by studying the cooperation and competition relationship between various agents.

[0003] Existing research on quadrotor formation control focuses on the impact of the outer loop (translation subsystem) on the inner loop (rotation subsystem), ignoring the two-way interaction between the inner and outer loops, which makes the control system less efficient and less coordinated in complex tasks and high dynamic environments. In addition, existing control strategies usually use complex nonlinear solution methods, such as adaptive dynamic programming methods, which are computationally intensive and difficult to implement in practical applications. At the same time, under the existing all-wheel drive system framework, the fault-tolerant control strategy is mainly used to ensure the stability of the system, and it is impossible to achieve the dual goals of system optimization and fault-tolerant control in the case of actuator failure. Therefore, it is urgent to design a fault-tolerant game mechanism and control strategy solution algorithm for the coordination of the inner and outer loops for the quadrotor cluster system, so as to reduce the computational complexity of the control strategy and improve the intelligence and safety of the quadrotor cluster system. Summary of the invention

[0004] Purpose of the invention: In view of the above shortcomings, the present invention provides a fault-tolerant game control method for a quadcopter cluster formation, which realizes the synchronous optimization of the inner and outer loops of attitude tracking and formation control, ensures that the cluster system can still maintain stability and optimal control performance under fault conditions, and finally realizes the interactive Steinberg equilibrium of the system under the coordination of the inner and outer loops.

[0005] Technical solution: To solve the above problems, the present invention discloses a fault-tolerant game control method for a cluster formation of quad-rotor helicopters, which specifically includes the following steps:

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

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

[0008] (3) Taking the translation subsystem of the quadrotor helicopter as the leader and the rotation subsystem as the follower, combined with the quadrotor helicopter cluster formation design, an inner and outer loop game model based on the incentive-based Steinberg differential graph game is constructed;

[0009] (4) Adopting the full-drive system method and introducing the fault-tolerant game controller, the controllers of the rotation subsystem and the translation subsystem are designed respectively, and the linear closed-loop rotation subsystem and the linear closed-loop translation subsystem of the quadrotor helicopter are constructed;

[0010] (5) Design the performance indicators of the linear closed-loop rotation subsystem of the quadrotor helicopter in the case of actuator failure; based on the fault estimation value, calculate the optimal fault-tolerant game control strategy of the linear closed-loop rotation subsystem of the quadrotor helicopter to achieve attitude tracking control of the quadrotor helicopters in the cluster in the case of actuator failure;

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

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

[0013] (1.1) Build a quadcopter cluster with n quadcopters. Set one quadcopter as the leader and the rest 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 quadcopter in the cluster, and an information interaction topology graph of the entire quadcopter cluster is constructed; wherein the information interaction topology graph within the cluster is defined as: Vertex Set represents the quadcopter node, and the edge set Represents the information interaction between quadrotors,Fig. is connected; in the vertex set In , the leader quadrotor is labeled as 0, and the remaining labels are assigned to the follower quadrotors;

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

[0016]

[0017] In the formula, v i is the control input of the quadcopter i; u i is the lift of the quadrotor helicopter i; is the Euler angle of the quadcopter i, φ i ,θ i ,ψ i are the roll angle, pitch angle and yaw angle of the quadrotor helicopter i respectively; is the inertia matrix of quadrotor helicopter i, are the moments of inertia of quadrotor helicopter i in the x, y, and z directions respectively; is the position of the quadcopter i, are the positions of the quadrotor helicopter i in the x, y, and z axes respectively; g is the acceleration due to gravity; is the control input matrix of the quadcopter i to the four rotors, is the parameter matrix of the quadrotor helicopter i, which is expressed as follows:

[0018]

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

[0020] (1.4) Construct the rotation subsystem and translation subsystem of the quadrotor helicopter i in the case of actuator failure:

[0021]

[0022] In the formula, is the effect of actuator failure of quadrotor helicopter i on the input of the rotation subsystem, is the impact of the actuator failure of the quadrotor helicopter i on the translation subsystem input, where:

[0023]

[0024] Where I4 is a 4×4 unit matrix, is the square of the rotation speed of the quadcopter i, ω i,1 ,ω i,2 ,ω i,3 ,ω i,4Indicates the rotation speed of the four rotors of the quadcopter i; is a diagonal matrix of fault values, where σ i,1 ,σ i,2 ,σ i,3 ,σ i,4 They represent the unknown fault levels of the four rotors of quadrotor helicopter i respectively; The specific expressions are as follows:

[0025]

[0026] In the formula,

[0027] Furthermore, the implementation process of designing the adaptive observer of the rotating subsystem of the quadrotor helicopter in step (2) is as follows:

[0028] Reconstruct the i-rotation subsystem of the quadrotor helicopter and obtain a system expression that is convenient for applying the adaptive observer method:

[0029]

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

[0031] Design an adaptive fault diagnosis observer for the reconfigured quadrotor helicopter rotation subsystem:

[0032]

[0033] In the formula, K i0 With K i1 is the feedback gain of the observer; and x i1 , x i2 and ρ i Estimated value of; 0 3×4 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] Formulate the dynamic equations for the state estimation error:

[0036]

[0037] In the formula, is the state estimation error, expressed as is the fault estimation deviation, expressed as K i With M i is the gain matrix, expressed as

[0038] Design the adaptive fault estimator for quadrotor helicopter i as follows:

[0039]

[0040] In the formula, Π i is a weight matrix that satisfies α i is a normal number, satisfying S i is a matrix that satisfies certain conditions: i >0 and Δ i is any number greater than 0;

[0041] The fault impact value of the rotation and translation subsystem is constructed by the obtained adaptive fault estimator and The estimated values ​​are:

[0042]

[0043] The following assumptions are made to ensure the convergence of the quadrotor helicopter fault estimator: there are two unknown constants and is i The absolute upper limit of is i The upper limit of the absolute value of the derivative, the estimated error satisfies the condition as well as

[0044] Under the assumed conditions, the fault estimator can ensure and Converge to the 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 ) is the matrix Δ in brackets i The minimum eigenvalue of max (S i ) is the matrix S in brackets i The maximum eigenvalue of The matrix in brackets The maximum eigenvalue of i1 ,b i2 ) has a value of b i1 , b i2 The smaller value; is max (S i ), The larger value.

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

[0050] The rotation subsystem of the quadrotor helicopter is used as the inner-loop follower to realize fault-tolerant attitude tracking control. The translation subsystem of the quadrotor helicopter is used as the outer-loop leader to realize fault-tolerant formation control. Combined with the setting of the leader helicopter and the follower helicopter in the quadrotor helicopter cluster formation and the mutual cooperation relationship between the helicopters in the cluster formation, an inner and outer loop game framework based on the incentive-type Steinberg differential graph game is constructed.

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

[0052] (4.1) Design the controller for the rotation subsystem of the quadrotor helicopter i:

[0053]

[0054] In the formula, ζ i A fault-tolerant game controller for the rotation subsystem;

[0055] Set the controller v i Bring in the rotation subsystem model and build a linear closed-loop rotation subsystem:

[0056]

[0057] Reconstruct the linear closed-loop rotation subsystem and obtain the following expression of the reconstructed system:

[0058]

[0059] Where ∈ is the reconstructed state variable, expressed as is the designed parameter matrix, expressed as 0 6i×3 is a zero matrix of size 6i×3, 0 6(n-i)×3 is a zero matrix of size 6(ni)×3; for The vector composed of the 0th-order derivative and the 1st-order derivative of is expressed as ∈( 0～1) is the vector of the 0th-order derivative and the 1st-order derivative of ∈, expressed as

[0060] (4.2) Design the controller for the translation subsystem of the quadrotor helicopter i:

[0061]

[0062] Where η i is the fault-tolerant game controller of the rotation subsystem, which has the following form:

[0063]

[0064] Where p is the designed state variable, expressed as p (0～1) is the vector of the 0th-order derivative and the 1st-order derivative of p, expressed as and is a parameter matrix that makes the policy u i Graph gaming is an acceptable strategy for the leader player in the formation;

[0065] Set the controller i Bring in the translation subsystem model and build a linear closed-loop translation subsystem:

[0066]

[0067] Reconstruct the linear closed-loop translation subsystem and obtain the following expression of the reconstructed system:

[0068]

[0069] In the formula, is the designed parameter matrix, expressed as for The vector composed of the 0th-order derivative and the 1st-order derivative of is expressed as

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

[0071] (5.1) The performance index of the inner ring rotation subsystem of the quadrotor helicopter i is designed as follows:

[0072]

[0073] In the formula, ∈ r is the desired tracking attitude angle, expressed as ∈ ir is the desired tracking attitude angle of quadrotor helicopter i; η i It is the fault-tolerant game controller of the outer-loop translation subsystem of the quadrotor helicopter i; Q i ,R i , Indicates performance index J i The weight matrix in , and there is Q i ≥0,R i >0,

[0074] (5.2) The fault-tolerant game control strategy model of the inner-ring rotation subsystem of the quadrotor helicopter i is designed as:

[0075]

[0076] The constraints are:

[0077]

[0078] (5.3) The optimal fault-tolerant game control strategy of the inner-ring rotation subsystem of the quadrotor helicopter i is calculated and obtained:

[0079]

[0080] Among them, λ i is the covariate variable, λ i satisfy:

[0081]

[0082] In the formula, is the designed parameter matrix,

[0083] (5.4) Convert the optimal fault-tolerant game control strategy of the inner-ring rotation subsystem of the quadrotor helicopter i into a state feedback form, and we get:

[0084]

[0085] Where P iis the solution of the asymmetric Riccati differential equation for the inner ring rotating subsystem of the quadrotor helicopter i, is the solution of the adjoint equation of the inner ring rotating subsystem of the quadrotor helicopter i, where the asymmetric Riccati equation and the adjoint equation are as follows:

[0086]

[0087] In the formula, is the designed parameter matrix of the quadrotor helicopter l, expressed as

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

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

[0090]

[0091] In the formula, p * is the vector composed of the target positions of all members of the quadcopter formation, expressed as is the desired tracking attitude angle of the quadrotor helicopter i; G i ,D i , Denotes the performance index Γ i The weight matrix in G i >0,D i >0; in the matrix In the cluster formation, for the leader, i=0, W0 is the set weight matrix, W0≥0, for followers in the cluster formation, i≠0,

[0092]

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

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

[0095]

[0096] The constraints are:

[0097]

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

[0099]

[0100] in, is a covariate variable that satisfies:

[0101]

[0102] In the formula, is the parameter matrix, expressed as

[0103] (6.4) Convert the optimal fault-tolerant game control strategy of the outer loop translation subsystem of the quadrotor helicopter i into a state feedback form, and we get:

[0104]

[0105] In the formula, ξ i For The Lagrange multiplier associated with Y i and [Y] i is the designed parameter matrix, is the solution of the asymmetric Riccati differential equation, is the solution of the adjoint equation, where the asymmetric Riccati equation and the adjoint equation are as follows:

[0106]

[0107] In the formula, Estimated impact of failure for all quadrotors and The vector formed, is the desired position p of the quadrotor helicopter * and the expected Euler angle ∈ r The vector formed is expressed as Φ, F, Satisfies the following differential equation:

[0108]

[0109] Where f is the fault impact value of all quadrotor helicopters and The vector formed is expressed as Ξ is the covariate variable λi , and the Lagrange multiplier β i The defined state quantity, where β i For The Lagrange multiplier involved; the differential equation is considered as a coupled system with two states as inputs to each other. is the state quantity The state matrix of The control input matrix, F is the state quantity For the fault distribution matrix of vector f, is the state matrix of the state quantity Ξ, is the control input matrix of the state quantity Ξ, For a system with state Ξ, for the vector The external input matrix.

[0110] Beneficial effects: Compared with the prior art, the beneficial effects of the present invention are as follows: First, the present invention reveals the bidirectional mutual influence mechanism between the inner-loop attitude tracking and the outer-loop formation control by constructing an inner-loop and outer-loop game framework based on the incentive-type Steinberg differential graph game; secondly, the designed adaptive fault observer realizes the real-time estimation of the actuator fault, and can dynamically adjust the control strategy when the system fails to ensure the stability of the system; finally, according to the fault estimation result of the adaptive observer, the fault-tolerant game control strategy of the inner and outer loops is designed; at the same time, based on the full-drive system theory, the fault-tolerant game strategy is expressed in the form of state feedback, and the solution of the Hamilton-Jacob equation is converted into the solution of the asymmetric Riccati differential equation, which significantly reduces the computational complexity and eliminates the dependence on the learning algorithm; the present invention overcomes the limitation of the prior art that only the unidirectional influence of the inner and outer loops is considered, effectively realizes the coordinated control of the inner and outer loops, enhances the fault tolerance of the system, and realizes the optimization of the synchronization performance of the inner and outer loops; the game strategy designed by the present invention can not only realize stable attitude tracking and formation control under fault conditions, but also realize interactive Steinberg equilibrium. BRIEF DESCRIPTION OF THE DRAWINGS

[0111] Figure 1 is a flow chart of the present invention;

[0112] Figure 2 This is a schematic diagram of a quad-rotor helicopter model;

[0113] Figure 3 This is the fault estimation error variation diagram under the incentive-based Steinberg differential graph game method;

[0114] Figure 4 It is the state estimation error variation diagram under the incentive-type Steinberg differential graph game method;

[0115] Figure 5 It is the trajectory diagram of the quadrotor helicopter cluster formation under the incentive-type Steinberg differential graph game method;

[0116] Figure 6 This is the attitude angle change diagram of the quadcopter cluster under the incentive-type Steinberg differential graph game method;

[0117] Figure 7 It is the control input variation diagram of the inner-ring rotation subsystem of the quadrotor helicopter cluster under the incentive-type Steinberg differential graph game method;

[0118] Figure 8 It is the control input variation diagram of the outer loop translation subsystem of the quadrotor helicopter cluster under the incentive-type Steinberg differential graph game method;

[0119] Fig. 9 This is the position change diagram of the quadcopter cluster under the incentive-type Steinberg differential graph game method;

[0120] Fig.10 Shown is a graph showing the change in attitude tracking error during the formation of a quadrotor helicopter cluster under the incentive-based Steinberg differential graph game method. DETAILED DESCRIPTION

[0121] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0122] like Figure 1 As shown, the present invention provides a fault-tolerant game control method for a quad-rotor helicopter cluster formation, which specifically includes the following steps:

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

[0124] Construct a quadrotor helicopter cluster with n quadrotor helicopters. Set one quadrotor helicopter as the leader in the cluster and set the remaining quadrotor helicopters as followers to form a formation with the leader.

[0125] Graph theory is used to describe the information interaction of each quadrotor in the cluster, so as to obtain the information interaction topology of the entire quadrotor cluster; among which, the information interaction topology within the cluster is defined as: Vertex Set represents the quadcopter node, and the edge set Represents the information interaction between quadrotors; in the vertex set In the graph, the leader quadrotor is labeled 0, and the remaining labels {1,…,n} are assigned to the follower nodes, and assuming that It’s China Unicom.

[0126] Considering the dual time scale characteristics of the inner and outer loops of the quadrotor helicopter, the rotation and translation subsystems of the quadrotor helicopter are established respectively. The rotation subsystem is:

[0127]

[0128] In the formula, is the Euler angle of the quadcopter i, φ i ,θ i ,ψ i are the roll angle, pitch angle and yaw angle of the quadrotor helicopter i respectively; is the inertia matrix of quadrotor helicopter i; is the input torque of the quadrotor helicopter i, expressed as:

[0129]

[0130] In the formula, γ i is the distance between the rotor and the center of mass of the quadrotor helicopter i; δ i is the drag coefficient of the quadrotor helicopter i; ε i is the positive proportional factor related to the propeller of the quadrotor helicopter i; are the control inputs of the roll angle, pitch angle and yaw angle of the quadrotor helicopter i, respectively, expressed as:

[0131]

[0132] In the formula, ω i,1 ,ω i,2 ,ω i,3 ,ω i,4 Indicates the rotation speed of the four rotors of the quadcopter i.

[0133] The control input of the Euler angle is distributed to the four rotors through the switchboard. By using the distribution matrix, the quadrotor helicopter rotation subsystem can be transformed into the following form:

[0134]

[0135] In the formula, the allocation matrix of quadrotor helicopter i is

[0136] By transforming the rotation subsystem of the quadcopter, the control quantity setting is simplified, so as to achieve the control target more efficiently. At the same time, in order to avoid the singular value phenomenon in the Euler angle expression, it is necessary to restrict the change of the Euler angle of the quadcopter. The restriction condition is: |φ i |<π / 2,|θ i |<π / 2,|ψi |<π / 2.

[0137] Quadcopter The translation subsystem is:

[0138]

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

[0140]

[0141] By redefining the control input, the quadrotor helicopter translation subsystem can be expressed as follows:

[0142]

[0143] In the formula, is the designed parameter matrix of quadrotor helicopter i, expressed as is the position of the quadcopter i, Where g is the acceleration due to gravity; u i is the redefined control input, which is expressed as follows:

[0144]

[0145] By transforming the translation subsystem of the quadrotor helicopter, the control quantity setting is simplified, so that the control target can be achieved more efficiently.

[0146] Considering the actuator failure that may occur during the flight of the quadcopter:

[0147]

[0148] In the formula, represents the square of the rotor rotation speed of the quadrotor drone i in the event of a fault; σ i,s To satisfy the condition 0<σ i,s ≤1 unknown fault degree; if the condition σ is satisfied i,s =1, then the quadcopter i is healthy; if the condition 0<σ i,s <1, then the quadrotor helicopter i has an actuator failure.

[0149] Considering the impact of actuator failure on the rotation and translation subsystems of the quadrotor helicopter, the subsystem models can be expressed as:

[0150]

[0151] In the formula, is the effect of actuator failure of quadrotor helicopter i on the input of the rotation subsystem, is the impact of the actuator failure of the quadrotor helicopter i on the translation subsystem input, expressed as:

[0152]

[0153] Where I4 is a 4×4 unit matrix, is the square of the rotation speed of the quadrotor helicopter i, where ω i,1 ,ω i,2 ,ω i,3 ,ω i,4 It represents the rotation speed of the four rotors of the quadcopter i, and the positions and rotation directions of the four rotors are as follows: Figure 2 As shown in; is a diagonal matrix of fault values, where σ i,1 ,σ i,2 ,σ i,3 ,σ i,4 They represent the unknown fault levels of the four rotors of the quadrotor drone i respectively; The specific expressions are as follows:

[0154]

[0155] In the formula,

[0156] By introducing the fault model of the quadrotor helicopter, the control method of the quadrotor helicopter under fault conditions can be studied.

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

[0158] Design an adaptive observer for the rotation subsystem of a quadrotor helicopter; reconstruct the rotation subsystem of the quadrotor helicopter and obtain a system expression that is convenient for applying the adaptive observer method.

[0159] By defining the matrix L i0 (v i )and It can be simplified and the new system expression is obtained:

[0160]

[0161] In the formula, the matrix defined is

[0162] Then redefine the system state quantity as well as The system is refactored to:

[0163]

[0164] In the formula, ρ i is the quantity related to the actuator failure, which can be expressed as I3 is the 3×3 identity matrix; 0 3×3 is a 3×3 zero matrix.

[0165] Design an adaptive fault diagnosis observer for the reconfigured quadrotor helicopter rotation subsystem:

[0166]

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

[0168] Based on the adaptive observer and the reconstructed translation subsystem, an adaptive fault estimator is designed; the dynamic equations about the state estimation deviation are constructed:

[0169]

[0170] In the formula, is the state estimation error, expressed as is the fault estimation deviation, expressed as K i With M i is the gain matrix, expressed as:

[0171] Design a fault estimator for the quadrotor helicopter as follows:

[0172]

[0173] In the formula, Π i is a weight matrix that satisfies α i is a normal number, satisfying S iis a matrix that satisfies certain conditions: i >0 and Δ i Any number greater than 0.

[0174] The fault impact value of the rotation and translation subsystem is constructed by the obtained adaptive fault estimator and The estimated values ​​are:

[0175]

[0176] It should be noted that in order to ensure the convergence of the quadrotor helicopter fault estimator, the following assumptions need to be made: there are two unknown constants and is i The absolute upper limit of is i The upper limit of the absolute value of the derivative, the estimated error satisfies the condition as well as

[0177] Under the assumed conditions, the fault estimator can ensure and Converge to the 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 ) is the matrix Δ in brackets i The minimum eigenvalue of max (S i ) is the matrix S in brackets i The maximum eigenvalue of The matrix in brackets The maximum eigenvalue of i1 ,b i2 ) has a value of b i1 , b i2 The smaller value; is max (S i ), The larger value.

[0182] Under the action of the adaptive observer, it can ensure that the state estimation deviation and the fault estimation deviation converge to a specific set Ensure the effectiveness of the fault-tolerant game controller introduced in step 4.

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

[0184] In the constructed incentive-based Steinberg differential graph game framework, the translation subsystem of the quadcopter acts as the leader player, and the rotation subsystem acts as the follower player. Specifically, the outer-loop translation subsystem announces the control strategy u to the inner-loop rotation 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 posture tracking control.

[0185] The follower player follows the control strategy u provided by the leader player i , make the best response to ensure that the posture tracking error is minimized and the graph game of the follower player reaches the Nash equilibrium. At the same time, in this process, by utilizing the fast time scale characteristics of the inner loop and establishing an adaptive fault estimator, the controllers for the inner and outer loops can be reconfigured in time, thereby improving the fault tolerance of the system.

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

[0187] Through game theory, on the one hand, the cooperative relationship between quadcopters is formulated as a graph game to achieve formation; on the other hand, active fault-tolerant control is achieved through the Steinberg game. And through the constructed incentive-based Steinberg differential graph game mechanism, the interaction between the outer ring leader and the inner ring follower in the quadcopter system is effectively optimized. At the same time, it should be noted that the leader player and the leader quadcopter are two different 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 player to form a complete game structure. In this case, the translation subsystem of all quadcopters acts as a leader in the incentive-based Steinberg differential graph game; the leader quadcopter is the physical component of the formation and is responsible for guiding the follower quadcopters to the desired destination.

[0188] Step 4: Adopting the all-wheel drive system method, introducing the fault-tolerant game controller, designing the controllers of the rotation subsystem and translation subsystem of the quadrotor helicopter respectively, and obtaining the linear closed-loop rotation subsystem and linear closed-loop translation subsystem of the quadrotor helicopter.

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

[0190] Design a controller for the rotating subsystem v i :

[0191]

[0192] In the formula, ζ i Fault-tolerant game controller for the rotation subsystem.

[0193] Set the controller v i Bring in the rotation subsystem model and build a linear closed-loop rotation subsystem:

[0194]

[0195] Reconstruct the linear closed-loop rotation subsystem and obtain the following expression of the reconstructed system:

[0196]

[0197] Where ∈ is the reconstructed state variable, expressed as is the designed parameter matrix, expressed as 0 6i×3 is a zero matrix of size 6i×3, 0 6(n-i)×3 is a zero matrix of size 6(ni)×3; for The vector composed of the 0th-order derivative and the 1st-order derivative of is expressed as ∈ (0～1) is the vector of the 0th-order derivative and the 1st-order derivative of ∈, expressed as

[0198] By introducing a linear closed-loop rotating subsystem, the subsequent performance index design and control quantity solution are made simpler and more efficient.

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

[0200]

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

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

[0203]

[0204] Reconstruct the linear closed-loop translation subsystem and obtain the following expression of the reconstructed system:

[0205]

[0206] In the formula, is the designed parameter matrix, expressed as p is the reconstructed state variable, expressed as p (0～1) is the vector of the 0th-order derivative and the 1st-order derivative of p, expressed as for The vector composed of the 0th-order derivative and the 1st-order derivative of is expressed as

[0207] By introducing a linear closed-loop translation subsystem, the subsequent performance index design and control quantity solution are made simpler and more efficient.

[0208] Step 5: Design the performance indicators of the quadrotor helicopter rotation subsystem in the cluster under the fault-tolerant game problem, and calculate the optimal fault-tolerant game control strategy of the quadrotor helicopter rotation subsystem based on the fault estimation value.

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

[0210]

[0211] In the formula, u xi ,u yi ,u zi They are the control inputs of the translation subsystem of the quadrotor helicopter in the x-axis, y-axis, and z-axis directions respectively; and is a parameter matrix that makes the policy u i Graph games are an acceptable strategy for the leader player in the formation.

[0212] Based on the control strategy i , calculate the thrust required by the inner ring rotating subsystem and the desired pitch angle θ ir and the roll angle φ ir :

[0213]

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

[0215]

[0216] In the formula, ∈ r is the desired tracking attitude angle, expressed as ∈ ir is the desired tracking attitude angle of the quadrotor helicopter i; Q i ,R i , Indicates performance index J i The weight matrix in , and there is Q i ≥0,R i >0,

[0217] Design a game control strategy model for the inner ring rotation subsystem. Specifically, the posture tracking problem of the inner ring rotation subsystem under actuator failure can be described as: given η i ,

[0218]

[0219] The constraints are:

[0220]

[0221] Based on the estimated faults obtained by the fault estimator, the optimal fault-tolerant game control strategy for the inner ring rotation subsystem of the quadrotor helicopter is calculated.

[0222] The Hamiltonian function of the rotating subsystem of helicopter i is defined as:

[0223]

[0224] Among them, λ i is the covariate variable of the i-th quadrotor helicopter rotation subsystem; optimal fault-tolerant game control strategy satisfy:

[0225]

[0226] Using the minimum principle, we can get:

[0227]

[0228] So we can get the optimal fault-tolerant game control strategy for the i-th quadrotor helicopter rotation subsystem for:

[0229]

[0230] Among them, λ i satisfy:

[0231]

[0232] In the formula,

[0233] The inner loop optimal fault-tolerant game control strategy Convert to state feedback expression. Set the co-state variable λ i In the following form:

[0234]

[0235] Where P i is the solution of the unsymmetric Riccati differential equation; is the solution of the adjoint equation; the asymmetric Riccati differential equation and the adjoint equation are as follows:

[0236]

[0237] In the formula,

[0238] The state feedback form of the fault-tolerant game control strategy of the inner ring rotation subsystem is obtained:

[0239]

[0240] In order to transform the fault-tolerant game strategy into a state feedback form, the adjoint equation is introduced, and then the estimated fault is used in the adjoint equation. Replace unknown fault Thus through Fault compensation is realized, the fault tolerance of the system is enhanced, and attitude tracking control is realized under the condition of inner loop fault. At the same time, the feedback matrix P i With fault compensation signal They are all coupled with the design of their neighboring quadrotors, reflecting the coordinated control between the quadrotors in the cluster and effectively improving the overall stability and fault tolerance of the system under actuator failure conditions.

[0241] Step 6: For the quadrotor helicopter cluster, design the performance indicators of the translation subsystems of the leader quadrotor helicopter and the follower quadrotor helicopter in the cluster formation under the fault-tolerant game problem and cluster formation task. At the same time, integrate the performance indicators of the translation subsystems of the leader quadrotor helicopter and the follower quadrotor helicopter in the cluster formation into one performance indicator, and calculate the optimal fault-tolerant game control strategy of the translation subsystem of the quadrotor helicopter in the cluster based on the fault estimation value.

[0242] Considering that the leader and follower in a quadcopter swarm have different goals, the performance indicators of the leader quadcopter and the follower quadcopter are designed separately. The leader quadcopter needs to optimize the trajectory to reach the desired destination, so its performance indicators are:

[0243]

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

[0245] At the same time, the follower quadrotor helicopter i is responsible for maintaining the formation shape, so its performance indicators are:

[0246]

[0247] In the formula, χ i0 are the relative positions of the i-th follower quadrotor helicopter and the leader quadrotor helicopter; il is the relative position of the i-th follower quadrotor helicopter and the j-th follower quadrotor helicopter; W i0 , W il , G i , D i is the performance index weight matrix of the i-th follower quadrotor helicopter, and W i0 ≥0,W il ≥0, G i >0,D i >0.

[0248] In order to ensure the consistency of the control strategy derivation of quadrotor helicopters in the cluster, the performance indicators of the leader quadrotor helicopter and the follower quadrotor helicopter in the formation are unified and integrated by redefining the system state quantity and weight matrix.

[0249] According to the desired destination and formation mode χ ij The numerical relationship of changes the target position of the follower quadrotor helicopter to the following form:

[0250]

[0251] So the performance index of the follower quadrotor helicopter can be written as:

[0252]

[0253] By resetting the system status The performance indicators of the leader and follower quadrotor helicopters are integrated into a unified form, and the performance indicators of the outer ring translation subsystem of the quadrotor helicopter i are designed as follows:

[0254]

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

[0256]

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

[0258]

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

[0260]

[0261] The constraints are:

[0262]

[0263] In the framework of the incentive Steinberg differential graph game, by introducing two constraints on the inner ring rotation subsystem, the leader fault-tolerant game control strategy η i The optimal response of the inner ring rotating subsystem is incorporated into the solution process and This enables a closed-loop design of the inner and outer loops, helping to improve the outer loop strategy’s ability to correct posture tracking errors.

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

[0265]

[0266] In the formula, is the covariate variable; i With ξ i is the Lagrange multiplier.

[0267] Optimal fault-tolerant game control strategy satisfy:

[0268]

[0269] Using the minimum principle, we can get:

[0270]

[0271] So we can get the optimal fault-tolerant game control strategy for:

[0272]

[0273] in β i ,ξ i satisfy:

[0274]

[0275] In order to enhance the fault tolerance of the quadrotor helicopter under actuator failure, the optimal fault-tolerant game control strategy is transformed It is a form of status feedback.

[0276] First, define the following state quantities:

[0277]

[0278] Then, through the defined state quantity, the following differential equation is derived:

[0279]

[0280] In the formula, the differential equation is regarded as a coupled system with two state quantities as inputs to each other, and f is the fault impact value of all quadrotor helicopters and The vector formed, is the desired position p of the quadrotor helicopter * and the expected Euler angle ∈ r The vector formed, is the state quantity The state matrix of The control input matrix, F is the state quantity For the fault distribution matrix of vector f, is the state matrix of the state quantity Ξ, is the control input matrix of the state quantity Ξ, For a system with state Ξ, for the vector The external input matrix is ​​as follows:

[0281]

[0282]

[0283] In the formula,

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

[0285]

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

[0287]

[0288] Satisfies the corresponding adjoint equation:

[0289]

[0290] In the formula, is the estimate of f, and is the estimated value of the failure effect of all quadrotors and The vectors are composed of

[0291] Therefore, the fault-tolerant game controller of the outer loop translation subsystem of the quadrotor helicopter i can be expressed in the following state feedback form:

[0292]

[0293] In the formula,

[0294] The fault-tolerant game strategy is converted into a state feedback form, the adjoint equation is introduced, and then the estimated fault is used in the adjoint equation. Replace the unknown fault f, so that Implement 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 quad-rotor helicopters in the cluster, thereby realizing the attitude tracking and formation control of all quad-rotor helicopters in the event of a fault when the formation cluster reaches a predetermined position.

[0296] This implementation uses a cluster consisting of three quadcopters to verify the effectiveness of the proposed fault-tolerant game control method, that is, the cluster point set

[0297] First, set the physical parameters, initial speed, initial angle, expected position and formation mode of the quadcopter. The simulation experiment of the quadcopter is carried out based on the following physical parameters:

[0298] m=2kg

[0299] g=9.81m / s 2

[0300]

[0301] Set the initial positions of the three quadrotor helicopters to:

[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 angles of the three quadcopters to 0:

[0306]

[0307] Set the desired destination of the quadcopter formation leader to:

[0308]

[0309] Set the formation mode of the quadcopter cluster to:

[0310] χ 10 = x 21 =[1,1,3] T

[0311] χ 20 =[2,2,6] T

[0312] Next, determine the time and severity of the quadcopter actuator failure. Set the quadcopter actuator failure to occur at the 10th second, and the severity of the failure is as follows:

[0313] Λ0=0.5I4

[0314] Λ1=0.6I4

[0315] Λ2=0.7I4

[0316] Then, set the weight matrix in the performance index, for the i-th drone, The weight matrix design is as follows:

[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 above set parameters, the simulation experiment started.

[0322] In order to show the correlation between the changes in simulation results and faults and fault-tolerant game strategies, the simulation process is divided into two stages according to the time when the fault occurs. The first stage is the simulation of the quadrotor helicopter cluster before the fault occurs; the second stage is the simulation of the quadrotor helicopter cluster after the fault occurs.

[0323] The first stage of the simulation experiment is: before the fault occurs, there is no actuator fault, so and is 0. Figure 5 The trajectory changes of the quadrotor helicopter cluster during the simulation are shown. Fig. 9 The coordinate changes of all helicopters in the quadcopter cluster during the simulation are shown. The simulation results show that, through the designed controller, the quadcopter has achieved the desired formation before the failure occurs, and each quadcopter has reached Fig. 9 The target location is shown in .

[0324] The second stage of simulation experiment is: the fault occurs at 10s. Figure 5 It can be seen that the quadcopter formation has different degrees of deviation. Figure 3 The difference between the estimated fault and the actual fault of the adaptive observer is shown in the simulation process. The simulation results show that the fault estimation deviation gradually converges to a neighborhood near zero, which shows the effectiveness of the fault estimator. Figure 4 The deviation between the estimated state quantity and the actual state quantity of the adaptive observer during the simulation process is shown. The simulation results in the figure show that the state estimation deviation gradually converges to a neighborhood near zero, which illustrates the effectiveness of the adaptive observer state estimation. Figure 6The simulation process shows the change of attitude angle of each quadrotor in the quadrotor cluster. The simulation results show that after the failure, the attitude angles of all quadrotors achieve smooth transition and finally remain stable, which shows the effectiveness of the fault-tolerant game controller. Figure 7 and Figure 8 The numerical changes of the inner and outer loops of the fault-tolerant game controllers designed for the quadcopter cluster during the simulation are shown respectively. The simulation results in the figure show that each quadcopter in the quadcopter cluster can maintain the stability of the fuselage attitude and hover at the desired position after completing the mission. Fig. 9 The simulation results in the figure show that each quadrotor helicopter in the quadrotor helicopter cluster can reach the desired position and achieve the formation goal. Fig.10 The change of attitude tracking error of the quadcopter cluster during the formation process is shown. The simulation results in the figure show that the attitude deviation of the quadcopter in the cluster continues to decrease and eventually converges to the same level, which illustrates the effectiveness of the inner loop attitude tracking control.

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

[0326] The above content has been described in detail the method of the present invention, but it is obvious that the specific implementation of the present invention is not limited thereto. For those skilled in the art, various obvious changes made to it without departing from the spirit of the present invention and the scope of the claims are within the protection scope of the present invention.

Claims

1. A fault-tolerant game control method for a quadrotor helicopter swarm formation, characterized in that: The following steps are involved: (1) Design a quadrotor helicopter swarm formation mode, construct an information interaction topology diagram of a quadrotor helicopter swarm, and establish the fault models of the rotation subsystem and translation subsystem of a single quadrotor helicopter in a quadrotor helicopter swarm; (2) Based on the single quadrotor fault model, an adaptive fault diagnosis observer for the quadrotor rotation subsystem is designed, and based on the reconstructed translation subsystem, an adaptive fault estimator is designed; (3) Taking the translation subsystem of the quadrotor helicopter as the leader and the rotation subsystem as the follower, combined with the quadrotor helicopter cluster formation design, an inner and outer loop game model based on the incentive-based Steinberg differential graph game is constructed; (4) Adopting the full-drive system method and introducing the fault-tolerant game controller, the controllers of the rotation subsystem and the translation subsystem are designed respectively, and the linear closed-loop rotation subsystem and the linear closed-loop translation subsystem of the quadrotor helicopter are constructed; (5) Design the performance indicators of the linear closed-loop rotation subsystem of the quadrotor helicopter in the case of actuator failure; based on the fault estimation value, calculate the optimal fault-tolerant game control strategy of the linear closed-loop rotation subsystem of the quadrotor helicopter to achieve attitude tracking control of the quadrotor helicopters in the cluster in the case of actuator failure; (6) Design performance indicators that can meet the requirements of quadrotor helicopter formation control in the event of actuator failure. At the same time, based on the fault estimation value obtained in step (2), calculate the optimal fault-tolerant game control strategy of the quadrotor helicopter translation subsystem to achieve cluster formation control of quadrotor helicopters in the event of actuator failure.

2. The fault-tolerant game control method for a quadrotor helicopter swarm formation according to claim 1, characterized in that: The implementation process of step (1) is as follows: (1.1) Build a quadcopter cluster with n quadcopters. Set one quadcopter as the leader and the rest as followers. The followers follow the leader to form a formation. (1.2) Graph theory is used to describe the information interaction of each quadcopter in the cluster, and an information interaction topology graph of the entire quadcopter cluster is constructed; wherein the information interaction topology graph within the cluster is defined as: Vertex Set represents the quadcopter node, and the edge set Represents the information interaction between quadrotors,Fig. is connected; in the vertex set In , the leader quadrotor is labeled as 0, and the remaining labels are assigned to the follower quadrotors; (1.3) The rotation and translation subsystems of the quadrotor helicopter i are established as follows: In the formula, v i is the control input of the quadcopter i; u i is the lift of the quadrotor helicopter i; is the Euler angle of the quadcopter i, φ i ,θ i , ψ i are the roll angle, pitch angle and yaw angle of the quadrotor helicopter i respectively; is the inertia matrix of quadrotor helicopter i, are the moments of inertia of quadrotor helicopter i in the x, y, and z axes respectively; is the position of the quadcopter i, are the positions of the quadrotor helicopter i in the x, y, and z axes respectively; g is the acceleration due to gravity; is the control input matrix of the quadcopter i to the four rotors, is the parameter matrix of the quadrotor helicopter i, which is expressed as follows: In the formula, γ i is the distance between the rotor and the center of mass of the quadrotor helicopter i; δ i is the drag coefficient of the quadrotor helicopter i; ε i is the propeller correlation coefficient of quadrotor helicopter i; m i is the mass of the quadrotor helicopter i; (1.4) Construct the rotation subsystem and translation subsystem of the quadrotor helicopter i in the case of actuator failure: In the formula, is the effect of actuator failure of quadrotor helicopter i on the input of the rotation subsystem, is the impact of the actuator failure of the quadrotor helicopter i on the translation subsystem input, where: Where I4 is a 4×4 unit matrix, is the square of the rotation speed of the quadcopter i, ω i,1 ,ω i,2 ,ω i,3 ,ω i，4 Indicates the rotation speed of the four rotors of the quadcopter i; is a diagonal matrix of fault values, where σ i，1 , σ i，2 , σ i，3 , σ i，4 They represent the unknown fault levels of the four rotors of quadrotor helicopter i respectively; The specific expressions are as follows: In the formula, 3. The fault-tolerant game control method for a quadrotor helicopter swarm formation according to claim 1, characterized in that: The implementation process of designing the adaptive observer of the rotating subsystem of the quadrotor helicopter in step (2) is as follows: Reconstruct the i-rotation subsystem of the quadrotor helicopter and obtain a system expression that is convenient for applying the adaptive observer method: In the formula, ρ i is the quantity related to the actuator failure, expressed as x i1 With x i2 is the designed state quantity, expressed as and The designed function mapping is expressed as I3 is a 3×3 identity matrix; 0 3×3 is a zero matrix of size 3×3; Design an adaptive fault diagnosis observer for the reconfigured quadrotor helicopter rotation subsystem: In the formula, K i0 With K i1 is the feedback gain of the observer; and x i1 , x i2 and ρ i Estimated value of; 0 3×4 is a zero matrix of size 3×4.

4. The fault-tolerant game control method for a quadrotor helicopter swarm formation according to claim 1, characterized in that: The implementation process of designing the adaptive fault estimator in step (2) is as follows: Formulate the dynamic equations for the state estimation error: In the formula, is the state estimation error, expressed as is the fault estimation deviation, expressed as K i With M i is the gain matrix, expressed as Design the adaptive fault estimator for quadrotor helicopter i as follows: In the formula, Π i is a weight matrix that satisfies α i is a normal number, satisfying S i is a matrix that satisfies certain conditions: i >0 and Δ i is any number greater than 0; The fault impact value of the rotation and translation subsystem is constructed by the obtained adaptive fault estimator and The estimated values ​​are: The following assumptions are made to ensure the convergence of the quadrotor helicopter fault estimator: There are two unknown constants and is i The absolute upper limit of is i The upper limit of the absolute value of the derivative, the estimated error satisfies the condition as well as Under the assumed conditions, the fault estimator can ensure and Converge to the set gather Defined as: In the formula, β i ,θ i The definition is as follows: In the formula, λ min (Δ i ) is the matrix Δ in brackets i The minimum eigenvalue of ; λ max (S i ) is the matrix S in brackets i The maximum eigenvalue of The matrix in brackets The maximum eigenvalue of min(b i1 , b i2 ) has a value of b i1 , b i2 The smaller value; is max (S i ), The larger value.

5. The fault-tolerant game control method for a quadrotor helicopter swarm formation according to claim 1, characterized in that: The implementation process of step (3) is as follows: The rotation subsystem of the quadrotor helicopter is used as the inner-loop follower to realize fault-tolerant attitude tracking control. The translation subsystem of the quadrotor helicopter is used as the outer-loop leader to realize fault-tolerant formation control. Combined with the setting of the leader helicopter and the follower helicopter in the quadrotor helicopter cluster formation and the mutual cooperation relationship between the helicopters in the cluster formation, an inner and outer loop game framework based on the incentive-type Steinberg differential graph game is constructed.

6. The fault-tolerant game control method for a quadcopter swarm formation according to claim 1, characterized in that: The implementation process of step (4) is as follows: (4.1) Design the controller for the rotation subsystem of the quadrotor helicopter i: In the formula, A fault-tolerant game controller for the rotation subsystem; Set the controller v i Bring in the rotation subsystem model and build a linear closed-loop rotation subsystem: Reconstruct the linear closed-loop rotation subsystem and obtain the following expression of the reconstructed system: Where ∈ is the reconstructed state variable, expressed as is the designed parameter matrix, expressed as 0 6i×3 is a zero matrix of size 6i×3, 0 6(n-i)×3 is a zero matrix of size 6(ni)×3; for The vector composed of the 0th-order derivative and the 1st-order derivative of is expressed as ∈ (0～1) is the vector of the 0th-order derivative and the 1st-order derivative of ∈, expressed as (4.2) Design the controller for the translation subsystem of the quadrotor helicopter i: Where η i is the fault-tolerant game controller of the rotation subsystem, which has the following form: Where p is the designed state variable, expressed as p (0～1) is the vector of the 0th-order derivative and the 1st-order derivative of p, expressed as and is a parameter matrix that makes the policy u i Graph gaming is an acceptable strategy for the leader player in the formation; Set the controller i Bring in the translation subsystem model and build a linear closed-loop translation subsystem: Reconstruct the linear closed-loop translation subsystem and obtain the following expression of the reconstructed system: In the formula, is the designed parameter matrix, expressed as for The vector composed of the 0th-order derivative and the 1st-order derivative of is expressed as 7. The fault-tolerant game control method for a quadcopter swarm formation according to claim 1, characterized in that: The implementation process of step (5) is as follows: (5.1) The performance index of the inner ring rotation subsystem of the quadrotor helicopter i is designed as follows: In the formula, ∈ r is the desired tracking attitude angle, expressed as ∈ ir is the desired tracking attitude angle of the quadrotor helicopter i; Q i , R i , Indicates performance index J i The weight matrix in , and there is (5.2) The fault-tolerant game control strategy model of the inner-ring rotation subsystem of the quadrotor helicopter i is designed as: The constraints are: (5.3) The optimal fault-tolerant game control strategy of the inner-ring rotation subsystem of the quadrotor helicopter i is calculated and obtained: Among them, λ i is the covariate variable, λ i satisfy: In the formula, is the designed parameter matrix, (5.4) Convert the optimal fault-tolerant game control strategy of the inner-ring rotation subsystem of the quadrotor helicopter i into a state feedback form, and we get: Where P i is the solution of the asymmetric Riccati differential equation for the inner ring rotating subsystem of the quadrotor helicopter i, is the solution of the adjoint equation of the inner ring rotating subsystem of the quadrotor helicopter i, where the asymmetric Riccati equation and the adjoint equation are as follows: In the formula, is the designed parameter matrix of the quadrotor helicopter l, expressed as 8. The fault-tolerant game control method for a quadcopter swarm formation according to claim 1, characterized in that: The implementation process of step (6) is as follows: (6.1) The performance index of the outer ring translation subsystem of the quadrotor helicopter i is designed as follows: Where p* is the vector composed of the target positions of all members of the quadrotor helicopter formation, expressed as is the desired tracking attitude angle of the quadrotor helicopter i; G i , D i , Denotes the performance index Γ i The weight matrix in G i >0,D i >0; in the matrix In the cluster formation, for the leader, i=0, W0 is the set weight matrix, W0≥0, for followers in the cluster formation, i≠0, In the formula, [W i ] nm (n∈(1, n+1), m∈(1, n+1)) is the set weight matrix; (6.2) The fault-tolerant game control strategy model of the translation subsystem of the quadrotor helicopter i is: The constraints are: (6.3) The optimal fault-tolerant game control strategy for the outer-loop translation subsystem of the quadrotor helicopter i is calculated and the formula is: in, is a covariate variable that satisfies: In the formula, is the parameter matrix, expressed as (6.4) Convert the optimal fault-tolerant game control strategy of the outer loop translation subsystem of the quadrotor helicopter i into a state feedback form, and we get: In the formula, ξ i For The Lagrange multiplier involved; Y i and [Y] i is the designed parameter matrix, is the solution of the asymmetric Riccati differential equation, is the solution of the adjoint equation, where the asymmetric Riccati equation and the adjoint equation are as follows: In the formula, Estimated impact of failure for all quadrotors and The vector formed, is the desired position p of the quadrotor helicopter * and the expected Euler angle ∈ r The vector formed is expressed as Φ, F, Satisfies the following differential equation: Where f is the fault impact value of all quadrotor helicopters and The vector formed is expressed as Ξ is the covariate variable λ i , and the Lagrange multiplier β i The defined state quantity, where β i For The Lagrange multiplier involved; the differential equation is considered as a coupled system with two states as inputs to each other. is the state quantity The state matrix of The control input matrix, F is the state quantity For the fault distribution matrix of vector f, is the state matrix of the state quantity Ξ, is the control input matrix of the state quantity Ξ, For a system with state Ξ, for the vector The external input matrix.

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