Aircraft formation control method based on fast adaptive nonsingular terminal sliding mode

Abstract

The invention discloses an aircraft formation control method based on a fast adaptive nonsingular terminal sliding mode, and the method comprises the steps: 1, considering the uncertainty and interference factors in a flight environment, and building a nonlinear dynamic model of a fixed-wing aircraft system; 2, analyzing information topology among the aircrafts by adopting graph theory knowledge, designing a distributed control architecture, and enabling each aircraft to establish a generalized formation tracking error based on the own state and the information of the adjacent aircrafts; step 3, designing a controller by using a nonsingular terminal sliding mode control theory so as to improve the accuracy and robustness of formation control and reduce a chattering phenomenon; and 4, in combination with a rapid adaptive control technology, adjusting control parameters in real time so as to adapt to external disturbance and system state change. By adopting a nonsingular terminal sliding mode control theory, the accuracy and robustness of formation control are improved, and the output buffeting phenomenon of the controller is reduced; and in combination with a rapid adaptive control technology, control parameters are adjusted in real time, and the stability and robustness of the system are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of fixed-wing aircraft formation control, and specifically designs a fixed-wing aircraft formation control method based on a distributed fast adaptive non-singular terminal sliding mode. Background Art

[0002] With the development of aircraft technology, fixed-wing aircraft are increasingly used in military and civilian fields. Fixed-wing aircraft clusters, due to their strong maneuverability and convenient deployment, can quickly respond to search and rescue missions, improve the accuracy of damage assessment and the effectiveness of decision-making, reduce casualties and property losses, and become an important part of new rescue equipment.

[0003] However, in the flight environment, the operating conditions are changeable, and there are many types of interference and uncertainties, including constant disturbances, such as gravity and slowly changing atmospheric disturbances, and dynamic disturbances, that is, aerodynamic drag related to the system state, which changes with the changes in the aircraft speed and attitude. These factors may affect the stability of the aircraft under different flight conditions.

[0004] Existing formation control methods mostly use centralized control to deal with uncertain interference, which has certain limitations when dealing with formation control in complex environments, especially the limited ability to deal with disturbances and uncertainties. Therefore, it is of great practical significance to study a new type of distributed formation control method. Summary of the invention

[0005] The purpose of the present invention is to provide a fixed-wing aircraft formation control method based on distributed fast adaptive non-singular terminal sliding mode, taking into account the uncertainty and interference factors in the flight environment, and establishing a nonlinear dynamic model of the fixed-wing aircraft system. Graph theory knowledge is used to analyze the information topology between aircraft, and a distributed control architecture is designed so that each aircraft establishes a generalized formation tracking error based on its own state and the information of neighboring aircraft. Using non-singular terminal sliding mode control theory, a controller is designed to improve the accuracy and robustness of formation control and reduce the jitter phenomenon. Combined with fast adaptive control technology, the control parameters are adjusted in real time to adapt to external disturbances and system state changes, effectively handling the uncertainty and disturbance in multi-aircraft formation flight, and improving the stability and robustness of the system.

[0006] The present invention discloses a fixed-wing aircraft formation control method based on a distributed fast adaptive non-singular terminal sliding mode, and the specific steps are as follows:

[0007] The steps include:

[0008] Step 1, considering the uncertainty and interference factors in the flight environment, establish a nonlinear dynamic model of the fixed-wing aircraft system;

[0009] Step 2: Use graph theory knowledge to analyze the information topology between aircraft and design a distributed control architecture so that each aircraft can establish a generalized formation tracking error based on its own state and the information of neighboring aircraft.

[0010] Step 3: Using the non-singular terminal sliding mode control theory, a controller is designed to improve the accuracy and robustness of formation control and reduce chattering.

[0011] Step 4: Combine fast adaptive control technology to adjust control parameters in real time to adapt to external disturbances and system state changes.

[0012] Preferably, in step 1, a nonlinear cluster system consisting of N aircraft is considered, and the motion model of each aircraft is as follows:

[0013]

[0014] Among them, p i =[x i ,y i , z i ] T represents the position of the virtual aircraft in the ground system, V represents the speed, θ represents the ballistic inclination, ψ vi Indicates the deflection angle.

[0015]

[0016] Among them, T i is the engine thrust, D i is the resistance, m i is the mass of the aircraft, g is the acceleration due to gravity, L i is the lift, φ i is the inclination angle. i 、Tilt angle φ i and lift L i As the control variable. Thrust T i Controlled by throttle, tilt angle φ i Controlled by the rudder and ailerons, the lift force L i Controlled by the elevator. The external disturbance is considered to be d iv , d iθ , It represents the external disturbance received by the system. The calculation of drag needs to consider the frontal area, flight speed, angle of attack and aerodynamic characteristics. In order to simplify the analysis and control design, drag is regarded as a nonlinear disturbance.

[0017] Furthermore, by defining the indirect variable Formula (26) can be transformed into

[0018]

[0019] Let v i =[V i ,θ i , ψ vi ] T , p i =[x i ,y i , z i ] T are the velocity vector and position vector of the i-th aircraft respectively. Taking the time derivative of equation (25), we obtain:

[0020]

[0021] in,

[0022]

[0023] Inserting equation (28) into equation (29), we can derive

[0024]

[0025] u i =[n iT , n iL cos(φ i ), n iL sin(φ i )] T

[0026]

[0027] And the matrix M i is reversible. In actual flight, the model parameters of fixed-wing aircraft may be affected by aerodynamic disturbances n iD The influence of parameter matrix N i Can be broken down into nominal parts and uncertainty

[0028]

[0029] The aircraft dynamics model can be further transformed into the following form:

[0030]

[0031] in, Represents multi-source composite disturbances, including external disturbances and internal parameter uncertainties.

[0032] Preferably, in step 2, a directed graph is used to describe the topological connectivity of the multi-aircraft communication network. The graph is represented as The node set v = {v1 , ..., v N} represents a group of N fixed-wing aircraft, and the edge set represents the information transmission between all neighboring aircraft. If (i, j)∈ε, it means that the information of the jth aircraft is available to the i-th aircraft. The neighborhood set of the i-th aircraft can be expressed as The adjacency matrix of a graph is represented by A = [a ij ]∈R N×N , if (i, j)∈ε, then a ij =1; otherwise, a ij = 0. In-degree matrix D in =diag{d 1 , ..., d N},in Then we can get the Laplace matrix L = D in -A.

[0033] Furthermore, based on the Laplace matrix L defined above, the normalized directed Laplace matrix can be defined as as follows:

[0034]

[0035] in,

[0036]

[0037] Furthermore, in this patent, we will focus on the formation control of fixed-wing aircraft based on the distributed leader-follower framework. The kinematic model of the aircraft platform is given by equation (32), which requires them to reach a given formation shape within a limited time and maintain coordinated motion with the leader. Based on the consistency consensus mechanism, the formation error variable can be defined as:

[0038] e ij =:p i -p j -E ij , i, j = 1, ..., N Formula (35)

[0039] Among them, p i and p j They represent the positions of the i-th and j-th aircraft at time t respectively; E ij represents the expected relative distance between the jth and ith aircraft at time t; e ij ∈R 3×1 ,t≤0 represents the position information error between the i-th and j-th aircraft. It is worth noting that e ij, i, j = L, 1, ..., N does not take into account the communication topology. In addition, for the i-th and j-th agents, a generalized error state is proposed:

[0040]

[0041] Preferably, in step 3, the controller is designed using non-singular terminal sliding mode control theory to improve the accuracy and robustness of formation control and reduce the buffeting phenomenon. Next, the overall design process adopts a novel sliding mode control method to ensure that the multi-aircraft system can achieve formation within a limited time. Consider the sliding mode variable, which is defined as follows:

[0042]

[0043] Among them, α i and β i is the control gain matrix, and α i =diag(α i1 , α i2 , α i3 ), β i =diag(β i1 ,β i2 ,β i3 ), and satisfies α in >0,β in >0, when n=1, 2, 3, and 1<γ<2. Sliding mode variable s i will reach zero for any initial value ∈ i (0) and Generalized formation tracking error ∈ i In a limited time t z The inner convergence is zero, satisfying the following inequality:

[0044]

[0045] Among them, η i =α i |∈ i |+β i |∈ i | γ On this basis, the equivalent control law is designed, and its formula is as follows:

[0046]

[0047] After the design of the non-singular terminal sliding mode controller is completed, in order to further improve the adaptability and stability of the system, the next step will be to combine fast adaptive control technology to adjust the control parameters in real time to ensure the system's rapid response and effective handling of external disturbances in a dynamic environment, thereby enhancing the overall performance of formation control.

[0048] Preferably, a fast adaptive control technology is used in step 4 to further optimize the control of the aircraft cluster formation. This technology adjusts the control parameters in real time to adapt to the disturbance of the external environment and the change of the internal state of the system, thereby enhancing the stability and robustness of the entire formation system.

[0049]

[0050] Among them, the control parameters and Positive number, 0<μ<1, adaptive gain and The update is based on the current and historical values ​​of the sliding mode variables and is adjusted through a specific adaptive law so that the control law can adapt to changes in system parameters and unknown external disturbances.

[0051]

[0052] Specifically, the control law consists of two parts: one is the equivalent control law based on the model The other is the arrival law control law used to ensure fast convergence The design of the arrival control law takes into account the sliding mode variable S i , and contains multiple positive control parameters and adaptive gains, which are dynamically updated according to the adaptive law.

[0053]

[0054] Finally, the control law u proposed in this patent i By combining the equivalent control law and the arrival control law A comprehensive control strategy was formed. This strategy not only improves the accuracy of aircraft formation control, but also significantly reduces the common buffeting phenomenon in the control system, making formation flight more stable and reliable.

[0055] Further, in step 4, considering the aggregate disturbance and uncertainty in the system, as well as the adaptive control law, we can define the adaptive estimation error, which includes and And construct the corresponding Lyapunov function.

[0056]

[0057] According to the previous derivation, the positive definiteness of the Lyapunov function is guaranteed. The derivative of the Lyapunov function expresses the change of the system state, and by calculating its rate of change, an expression for the change of the system energy is obtained.

[0058]

[0059] Further deduction

[0060]

[0061] To facilitate subsequent derivation, the system state is redefined as the relevant state variable, and the stability analysis is made more intuitive. By analyzing the derivative of the Lyapunov function, the relationship between it and the sliding mode variable and the adaptive gain can be found, so as to further derive the conditions for system stability.

[0062]

[0063] Finally, by defining some symbols to make the expression more concise, the relationship between the rate of change of the Lyapunov function and the system state is obtained. Based on the defined parameters and symbols, it can be proved that under the proposed control law, the Lyapunov function will converge to zero from any initial condition and meet the finite-time stability criterion.

[0064]

[0065] Therefore, the final conclusion is that under the proposed formation control law, the system will be able to reach a stable state within a finite time, and the proof process is completed.

[0066] Compared with the prior art, the present invention has at least one of the following beneficial effects:

[0067] (1) The present invention creatively proposes an aircraft formation control method based on fast adaptive non-singular terminal sliding mode, which improves the accuracy and robustness of formation control and reduces controller output chattering by adopting non-singular terminal sliding mode control theory;

[0068] (2) The present invention combines fast adaptive control technology to adjust control parameters in real time to adapt to external disturbances and system state changes, effectively handle uncertainties and disturbances in multi-aircraft formation flight, and improve the stability and robustness of the system;

[0069] (3) The present invention uses graph theory knowledge to analyze the information topology between aircraft and designs a distributed control architecture, so that each aircraft can establish a generalized formation tracking error based on its own state and the information of neighboring aircraft, thereby reducing dependence on the control center and improving system scalability. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 It is a schematic diagram of the flow chart of the fixed-wing aircraft formation control method of the present invention;

[0071] Figure 2Schematic diagram of the fixed-wing aircraft formation control strategy program framework of the present invention

[0072] Figure 3 This is a schematic diagram of the fixed-wing aircraft motion model of the present invention;

[0073] Figure 4 This is a schematic diagram of the fixed-wing aircraft formation communication topology of the present invention;

[0074] Figure 5 This is a schematic diagram of the fixed-wing aircraft formation position design of the present invention;

[0075] Figure 6 This is a schematic diagram showing the fixed-wing aircraft formation error of the present invention;

[0076] Figure 7 This is a schematic diagram of the three-axis control input of the fixed-wing aircraft formation of the present invention;

[0077] Figure 8 It is a schematic diagram of the three-dimensional formation flight trajectory of the fixed-wing aircraft formation of the present invention. DETAILED DESCRIPTION

[0078] In one embodiment of the present invention, a fixed-wing aircraft formation control method based on a distributed fast non-singular terminal sliding mode is provided. Figure 1-8 The specific implementation of the present invention is described in detail. The aircraft formation control method first uses Figure 1 The task is planned by the flowchart shown in the figure, and then Figure 2 The control strategy and feedback information system framework of aircraft clusters are established and implemented. Figure 3 Supported by the motion model, combined with Figure 4 The communication topology characterization and Figure 5 The time-varying formation position design is realized by Figure 6 Collect and analyze formation error information and Figure 7 The three-axis control input shown ensures precise control of each aircraft and ultimately achieves Figure 8 The three-dimensional formation flight trajectory is shown.

[0079] Figure 1 The process of the control method of the present invention is shown. The process starts from startup, including steps such as establishing the aircraft's motion model, communication topology, and designing the control strategy, until the formation control task is completed. Figure 2 The program framework of the control strategy of the present invention is shown. The program starts with initializing the state of the aircraft cluster, including information such as position, speed and attitude. Then, the program enters the main loop, obtains the sensor data of each aircraft in real time, and calculates the control input according to the distributed control law. The control input includes thrust, tilt angle and lift, which are used to adjust the motion state of the aircraft to maintain the desired formation shape.

[0080] The present invention comprises the following steps:

[0081] Step 1: Considering the uncertainty and interference factors in the flight environment, a nonlinear dynamic model of the fixed-wing aircraft system is established.

[0082] Figure 3 The motion model of the fixed-wing aircraft in the present invention is described. The model takes into account the dynamic characteristics of the aircraft. By establishing a nonlinear dynamic model, the present invention can accurately grasp the flight state of each aircraft.

[0083] Step 2: Use graph theory knowledge to analyze the information topology between aircraft and design a distributed control architecture so that each aircraft can establish a generalized formation tracking error based on its own state and the information of neighboring aircraft.

[0084] Figure 4 The communication topology of a fixed-wing aircraft formation network is presented. This topology illustrates how information is exchanged between drones and is key to implementing distributed control strategies.

[0085] Figure 5 It describes the predetermined position and relative relationship of each aircraft in the formation. These position designs are the basis for achieving specific formation tasks.

[0086] Step 3: Using the non-singular terminal sliding mode control theory, design a controller to improve the accuracy and robustness of formation control and alleviate the chattering phenomenon.

[0087] Step 4: Combine fast adaptive control technology to adjust control parameters in real time to adapt to external disturbances and system state changes, thereby improving system stability and robustness.

[0088] Figure 6 The performance of formation error in the process of performing formation flight mission is described. By designing appropriate control law and adaptive law, the present invention can effectively reduce formation error and ensure the accuracy and stability of formation flight.

[0089] Figure 7 The three-axis control input of the aircraft in the present invention is shown. The control input includes rolling, pitching and yaw control along the aircraft coordinate system, and these control inputs are used to adjust the attitude and trajectory of the aircraft to meet the requirements of formation flying.

[0090] Figure 8 The three-dimensional trajectory of multiple fixed-wing aircraft when performing a formation flight mission is shown. Through the distributed control strategy, the aircraft can fly in coordination according to the predetermined trajectory and maintain a stable formation structure. This figure illustrates the ability of the present invention to maintain a formation in a complex space environment.

[0091] Through the above implementation modes, the present invention not only improves the accuracy and robustness of fixed-wing aircraft formation control, but also enhances the stability and adaptability of the entire system through a distributed control strategy.

[0092] The following specific embodiments describe in detail the initial conditions and preset trajectories of the simulation, the formation topology and formation position design in a multi-fixed-wing aircraft system to verify the effectiveness of the proposed control method.

[0093] We consider a multi-fixed-wing aircraft formation consisting of five UAVs (i.e., N = 5), the pilot UAV, i.e., the UAV 0 , the predetermined trajectory is defined as p 0 =[x 0 ,y 0 , z 0 ] T , the specific equations are as follows,

[0094]

[0095] At the beginning of the simulation, the initial positions of the four following drones are set to: 1 (0) = [-5, 10, 15] T , p 2 (0) = 15, 20, 20] T , p 3 (0) = [20, 15, -10] T , p 4 (0) = [-15, 25, -15] T The initial velocity of all drones is the same, and the initial velocity vector and attitude angle [v i (0), θ i (0), ψ vi (0)] T =[10,0,0] T , i = 0, ..., 4.

[0096] The multi-UAV system of the present invention is composed of a leader (denoted by subscript 0) and four followers (denoted by subscripts 1, 2, 3, and 4, respectively). Figure 4 As shown in Figure 1, the communication topology is the basis for implementing distributed control strategies, which describes in detail how information is exchanged between drones. In this topology, each drone can communicate with its neighbors, ensuring real-time sharing of information and rapid delivery of instructions.

[0097] The adjacency matrix A describes the communication connection between drones, where the matrix element aij indicates whether drone i can directly receive information from drone j, specifically:

[0098]

[0099] The Laplace matrix L is used to analyze the stability and consistency of the entire formation. It is obtained by calculating the difference between the adjacency matrix and the degree matrix. Specifically,

[0100]

[0101] In addition, in order to further improve the adaptability and flexibility of the control strategy, we also designed the normalized Laplace matrix This matrix is ​​normalized by dividing each element of the Laplacian matrix by the maximum degree in the network, making the control strategy more robust to changes in the network size.

[0102]

[0103] Furthermore, in this embodiment, the position of the leader is set as the center of the entire formation, and the followers are arranged around the leader according to a predetermined formation design. The expected relative position between each follower drone and its leader also specifically defines the expected formation position E ij , which means that the follower p j With the leader p i The relative distance between them is defined by the predefined distance d i0 This design method allows us to accurately calculate the position that each follower drone should maintain in three-dimensional space, thereby forming a stable and controllable formation structure. The specific formula is as follows:

[0104]

[0105] Among them, the relative position in the local coordinate system is X 10 =-d 10 sin(Φ),Y 10 =d 10 cos(Φ),X 20 =-d 20 sin(Φ),Y 20 =-d 20 cos(Φ),X 30 =-d 30 sin(Φ),Y 30 =d 30 cos(Φ),X 40 =-d 40 sin(Φ),Y 40 =-d 40 cos(Φ), and the rotation matrix

[0106] Through this formation topology and array position design, the present invention not only improves the precision and robustness of fixed-wing aircraft formation control, but also enhances the stability and adaptability of the entire system through distributed control strategies. These designs provide a solid foundation for realizing complex formation tasks.

[0107] Finally, in the implementation of the present invention, the selection of control parameters is crucial to achieve the desired tracking performance and overall stability of the system. The following is a design guide for control parameters and their specific values ​​for the proposed adaptive fast non-singular terminal sliding mode control (DFANTSM) method. The relevant formulas can be referred to above. For ease of reading, some key formulas are expressed as follows:

[0108]

[0109] To ensure the stability of the system in the face of disturbances, the control gain The setting of these parameters aims to balance the response speed of the system and the smoothness of the control signal. i =diag(2,2,2),β i =diag(1,1,1) and Affects the convergence speed and amplitude of the formation error. Adaptive gain It has an important influence on the convergence rate of the adaptive estimation error, and its value can effectively reduce the adaptation time while avoiding saturation and overshoot of the control input.

[0110] Through the design of the above control parameters, the present invention can achieve efficient formation control in a multi-fixed-wing aircraft system. The reasonable selection of these parameters ensures the stability and robustness of the system in the face of uncertainty and external disturbances, providing a solid foundation for achieving precise formation flight.

Claims

1. A method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode, characterized in that: The steps include: Step 1, considering the uncertainty and interference factors in the flight environment, establish a nonlinear dynamic model of the fixed-wing aircraft system; Step 2: Use graph theory knowledge to analyze the information topology between aircraft and design a distributed control architecture so that each aircraft can establish a generalized formation tracking error based on its own state and the information of neighboring aircraft. Step 3: Using the non-singular terminal sliding mode control theory, a controller is designed to improve the accuracy and robustness of formation control and reduce chattering. Step 4: Combine fast adaptive control technology to adjust control parameters in real time to adapt to external disturbances and system state changes.

2. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 1, characterized in that: In step 1: Consider a nonlinear cluster system consisting of N aircraft. The motion model of each aircraft is as follows: Among them, p i =[x i ,y i , z i ] T represents the position of the virtual aircraft in the ground system, V represents the speed, θ represents the ballistic inclination, ψ vi Indicates the deflection angle; Among them, T i is the engine thrust, D i is the resistance, m i is the mass of the aircraft, g is the acceleration due to gravity, L i is the lift, φ i is the inclination angle; the thrust T i 、Tilt angle φ i and lift L i As the control variable; thrust T i Controlled by throttle, tilt angle φ i Controlled by the rudder and ailerons, the lift force L i Controlled by elevator; external disturbance is considered as d iv , d iθ It represents the external disturbance received by the system. The calculation of drag needs to consider the frontal area, flight speed, angle of attack and aerodynamic characteristics. In order to simplify the analysis and control design, the drag is regarded as a nonlinear disturbance.

3. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 2, characterized in that: By defining indirect variables Formula (26) can be transformed into Let v i =[V i ,θ i , ψ vi ] T , p i =[x i ,y i , z i ] T are the velocity vector and position vector of the i-th aircraft respectively. Taking the time derivative of equation (25), we obtain: in, Inserting equation (28) into equation (29), we can derive you i =[n iT ,n iL cos(φ i ),n iL sin(φ i )] T And the matrix M i It is reversible.

4. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 3, characterized in that: In actual flight, the model parameters of fixed-wing aircraft may be affected by aerodynamic disturbances. iD The influence of parameter matrix N i Can be broken down into nominal parts and uncertainty The aircraft dynamics model can be further transformed into the following form: in, Represents multi-source composite disturbances, including external disturbances and internal parameter uncertainties.

5. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 4, characterized in that: In step 2, a directed graph is used to describe the topological connectivity of the multi-aircraft communication network; the graph is represented by G = (v, ε}, where the node set v = {v1, ..., v N } represents a group of N fixed-wing aircraft, and the edge set represents the information transmission between all neighboring aircraft; if (i, j)∈ε, it means that the information of the jth aircraft is available to the i-th aircraft; the neighborhood set of the i-th aircraft can be expressed as The adjacency matrix of a graph is represented by A = [a ij ]∈R N×N , if (i, j)∈ε, then a ij =1; otherwise , a ij =0; in-degree matrix D in =diag{d1,...,d N },in Then we can get the Laplace matrix L = D in -A.

6. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 5, characterized in that: Based on the Laplace matrix L defined above, the normalized directed Laplace matrix can be defined as as follows: in, The focus is on the fixed-wing aircraft formation control based on the distributed leader-follower framework. The kinematic model of the aircraft platform is given by equation (32), which requires them to reach a given formation shape within a limited time and maintain coordinated motion with the leader. Based on the consistency consensus mechanism, the formation error variable can be defined as: e ij :=p i -p j -E ij , i, j = 1, ..., N Formula (11) Among them, p i and p j They represent the positions of the i-th and j-th aircraft at time t respectively; E ij represents the expected relative distance between the jth and ith aircraft at time t; e ij ∈R 3×1 ,t≤0 represents the position information error between the i-th and j-th aircraft; it is worth noting that e ij , i, j = L, 1, ..., N is defined without considering the communication topology; in addition, a generalized error state is proposed for the i-th and j-th agents:

7. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 6, characterized in that: The overall design process in step 3 adopts a sliding mode control method to ensure that the multi-aircraft system can form a formation within a limited time; considering the sliding mode variables, they are defined as follows: Among them, α i and β i is the control gain matrix, and α i =diag(α i1 , α i2 , α i3 ), β i =diag(β i1 ,β i2 ,β i3 ), and satisfies α in >0,β in >0, when n=1, 2, 3, and 1<γ<2; sliding mode variable S i will reach zero for any initial value ∈ i (0) and Generalized formation tracking error ∈ i In a limited time t z The inner convergence is zero, satisfying the following inequality: Among them, η i =α i |∈ i |+β i |∈ i | γ On this basis, the equivalent control law is designed, and its formula is as follows: After the design of the non-singular terminal sliding mode controller is completed, in order to further improve the adaptability and stability of the system, the next step will be to combine fast adaptive control technology to adjust the control parameters in real time to ensure the system's rapid response and effective handling of external disturbances in a dynamic environment, thereby enhancing the overall performance of formation control.

8. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 7, characterized in that: In step 4, the control parameters are adjusted in real time to adapt to the disturbance of the external environment and the change of the internal state of the system, thereby enhancing the stability and robustness of the entire formation system; say(S i )=[say(s 1i (say) 2i (say) 3i )] T sgn(S i )=[sgn(s 1i )sgn(s 2i )sgn(s 3i )] T n=1, 2, 3, i=1,...,N Among them, the control parameters and Positive number, 0<μ<1, adaptive gain and The update of is based on the current value and historical value of the sliding mode variable, and is adjusted through a specific adaptive law, so that the control law can adapt to changes in system parameters and unknown external disturbances; 9. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 8, characterized in that: The control law consists of two parts: one is the equivalent control law based on the model The other is the arrival law control law used to ensure fast convergence The design of the arrival control law takes into account the sliding mode variable S i , and contains multiple positive control parameters and adaptive gains, which are dynamically updated according to the adaptive law; Finally, the control law u i By combining the equivalent control law and the arrival control law A comprehensive control strategy is formed.

10. The method for controlling an aircraft formation based on a fast adaptive non-singular terminal sliding mode according to claim 9, characterized in that: Considering the aggregate disturbance and uncertainty in the system, as well as the adaptive control law, the adaptive estimation error is defined. The adaptive estimation error includes and And construct the corresponding Lyapunov function; According to the previous derivation, the positive definiteness of the Lyapunov function is guaranteed; the derivative of the Lyapunov function expresses the change of the system state, and by calculating its rate of change, an expression for the change of the system energy is obtained; Further deduction To facilitate subsequent derivation, the system state is redefined as the relevant state variable, making the stability analysis more intuitive; by analyzing the derivative of the Lyapunov function, the relationship between it and the sliding mode variable and the adaptive gain can be found, thus further deducing the conditions for system stability; Finally, by defining some symbols to make the expression more concise, the relationship between the rate of change of the Lyapunov function and the system state is obtained; according to the defined parameters and symbols, it is proved that under the proposed control law, the Lyapunov function will converge to zero from any initial condition and meet the finite-time stability criterion; Therefore, the final conclusion is that under the proposed formation control law, the system will be able to reach a stable state within a finite time, and the proof process is completed.

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