Multi-uav fixed-time dynamic event-triggered output feedback control method

Abstract

The present application relates to a kind of multi-unmanned plane fixed time dynamic event trigger output feedback control method, determine the communication topology structure of multiple six rotor unmanned plane;The attitude physical characteristics of single six rotor unmanned plane are modeled;Obtain the state equation of the attitude system of the i follower unmanned plane;For the i follower unmanned plane, design dynamic event trigger mechanism, establish fuzzy logic state observer, estimate the unmeasurable state of system, obtain the estimated value of system state;According to the estimated value of system state and the desired Euler angle trajectory of virtual leader, the consistency tracking error of the i follower unmanned plane system is constructed;Based on consistency tracking error and dynamic event trigger mechanism, under the framework of backstepping technique, actual controller and uncertain parameter adaptive law are constructed;Based on actual controller, all follower unmanned plane tracks the desired Euler angle trajectory of virtual leader.This technical solution reduces the communication burden and saves communication resources.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of unmanned aerial vehicle control, and particularly relates to a multi-unmanned aerial vehicle fixed-time dynamic event-triggered output feedback control method. BACKGROUND

[0002] Multi-rotor unmanned aerial vehicles have been widely used in geographic surveying, biological monitoring and military monitoring fields due to their good maneuverability, superior hovering and low-speed flight performance. Multi-rotor unmanned aerial vehicles have the characteristics of simple structure, flexible control, strong maneuverability, vertical take-off and landing, and flight safety. As a high-precision nonlinear system, it is a challenging task to maintain the stability of the attitude system due to the problems of insufficient driving, strong nonlinearity and coupling of unmanned aerial vehicles.

[0003] In reality, the states of unmanned aerial vehicles are not all measurable, and how to deal with these unmeasurable states has become a difficult problem in the design of unmanned aerial vehicle controllers.

[0004] The state observer, as a commonly used method to deal with unmeasurable states, has been widely applied, such as documents 【1】-【5】:

[0005] Document 【1】: Chen, C. L. P., Wen, G., Liu, Y., Liu, Z.: Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Transactions on Cybernetics. 46(7): 1591-1601 (2016).

[0006] Document 【2】: Xiao, W., Cao, L., Li, H., Lu, R.: Observer-based adaptive consensus control for nonlinear multi-agent systems with time-delay. Science China Information Sciences. 63: 132202 (2020).

[0007] Document [3]: Chen, L., Tong, S.: Observer-based adaptive fuzzy consensus control of nonlinear multi-agent systems encountering deception attacks. IEEE Transactions on Industrial Informatics. 20(2): 1808-1818 (2024).

[0008] Document [4]: Wang, L., Chen, C. L. P.: Reduced-order observer-based dynamic event-triggered adaptive NN control for stochastic nonlinear systems subject to unknown input saturation. IEEE Transactions on Neural Networks and Learning Systems. 32(4): 1678-1690 (2021).

[0009] Document [5]: Lin, G., Li, H., Ahn, C. K., Yao, D.: Event-based finite-time neural control for human-in-the-loop UAV attitude systems. IEEE Transactions on Neural Networks and Learning Systems. 34(12): 10387-10397 (2023).

[0010] Document [6]: Zhao, Y., Tang, F., Zong, G., Zhao, X., Xu, N.: Event-based adaptive containment control for nonlinear multiagent systems with periodic disturbances. IEEE Transactions on Circuits and Systems II: Express Briefs. 69(12): 5049-5053 (2022).

[0011] In document 【1】, an adaptive output feedback consensus tracking controller is designed for high-order multi-agent systems. In document 【2】, an adaptive consensus tracking controller based on an observer is designed for multi-agent systems with input time-delay and input saturation. In document 【3】, an adaptive fuzzy consensus controller based on an observer is designed for multi-agent systems with network communication encountering deception attacks. In document 【4】, an adaptive neural network controller based on a reduced-order observer is designed for nonlinear stochastic systems with unknown input saturation based on a dynamic event-triggered mechanism. In document 【5】, an event-triggered finite-time attitude controller is designed for human-in-the-loop unmanned aerial vehicle systems with disturbances. In document 【6】, an event-triggered controller is designed for nonlinear multi-agent systems with periodic disturbances.

[0012] The above prior art has the following technical problems:

[0013] 1. In documents 【1】-【3】, the communication burden and communication resources are not considered. Since it is inevitable to reduce the communication burden and save the communication resources in real life, this situation is too idealistic, and these methods cannot be applied to actual multi-unmanned aerial vehicle systems.

[0014] 2. In document 【4】, although the proposed controller saves communication resources and reduces communication burden, it cannot achieve good instantaneous performance of the system. For actual multi-unmanned aerial vehicle systems, due to the existence of various uncertain factors in the environment, the designed controller often needs to achieve good instantaneous performance, so this method is not suitable for actual multi-unmanned aerial vehicle systems.

[0015] 3. In document 【5】, the designed controller can only achieve finite-time control, which makes the settling time always change with the initial state, and it is not feasible for the case where the initial state is unknown. In practice, for multi-unmanned aerial vehicle systems, fast convergence independent of the initial state is essential. Therefore, this method is also not suitable for actual unmanned aerial vehicle systems.

[0016] 4. In document 【6】, the inclusion controller designed based on a dynamic event-triggered mechanism saves communication resources, but the number of triggers is still large, and it does not achieve the purpose of maintaining high transmission efficiency of the controller and reducing the communication burden.

[0017] As a common actual system, how to save communication resources while there are unmeasurable states and achieve fixed-time convergence control of unmanned aerial vehicles is a problem that needs to be solved in the field of unmanned aerial vehicle control. SUMMARY

[0018] The technical problem to be solved by the present invention is to overcome the shortcomings of the prior art and provide a method for triggering output feedback control of dynamic events at fixed time for multiple UAVs.

[0019] To solve the above-mentioned technical problems, the technical solution of the present invention is as follows:

[0020] A method for output feedback control of multiple unmanned aerial vehicles (UAVs) triggered by dynamic events at fixed times includes the following steps:

[0021] Step 1: Determine the communication topology of multiple hexacopter UAVs;

[0022] Step 2: Model the attitude physical characteristics of a single hexacopter UAV to obtain the attitude system model of a single hexacopter UAV;

[0023] Step 3: Obtain the state equation of the attitude system of the i-th follower UAV based on the attitude system model of a single hexacopter UAV;

[0024] Step 4: Design a dynamic event triggering mechanism for the i-th follower drone;

[0025] Step 5: For the i-th follower drone, establish a fuzzy logic state observer to estimate the unmeasurable state of the system and obtain the estimated value of the system state;

[0026] Step 6: Construct the consistency tracking error of the i-th follower drone system based on the estimated system state and the expected Euler angle trajectory of the virtual leader;

[0027] Step 7: Based on the consistency tracking error and dynamic event triggering mechanism, and within the framework of the backstepping technique, first construct the Lyapunov function V. i,1 According to the Lyapunov function V i,1 Building a virtual controller α i,1 Adaptive Law of Uncertain Parameters Reconstruct the Lyapunov function V i,2 According to the Lyapunov function V i,2 Building an actual controller ν i (t) and the adaptive law of uncertain parameters

[0028] Step 8: Based on the actual controller ν i (t) Control all follower drones to track the virtual leader's desired Euler angle trajectory.

[0029] Furthermore, in step one, the communication topology of the multiple hexacopter drones is as follows: the communication relationship between the N follower drones is represented by an undirected graph. To describe, in order to express the information exchange relationship, where, denotes a set of N follower drones, denotes a set of edges, denotes an adjacency matrix with non-negative adjacency elements; a i,k denotes the dependence of the i-th follower drone on the k-th follower drone, i = 1, 2,..., N; the set of neighbors of the i-th follower drone n i is denoted as If the i-th follower drone can directly obtain information from the k-th follower drone, i.e., the k-th follower drone a i,k > 0, otherwise, a i,k = 0; for an undirected graph, a i,k = a k,i , a k,i denotes the dependence of the k-th follower drone on the i-th follower drone; the in-degree matrix is characterized by where the Laplacian matrix of an undirected graph is denoted as The leader adjacency matrix of an undirected graph is defined as where a i,0 denotes the dependence of the i-th follower drone on the leader; if the i-th drone can directly obtain information from the leader, then a i,0 > 0, otherwise, a i,0 = 0.

[0030] Further, in step two, the single hexacopter drone attitude system model is:

[0031]

[0032] where is a transformation matrix, is an inertia matrix, is a gyroscopic torque, and E is the total torque;

[0033] φ = [φ1, φ2, φ3] T , φ1is the roll angle, φ2is the pitch angle, and φ3is the yaw angle;

[0034]

[0035] ω = [ω x , ω y , ω z ] T , ω x is the roll angular velocity, ω y is the pitch angular velocity, and ω z is the yaw angular velocity;

[0036]

[0037] is the gyroscopic torque of the roll angle, is the gyroscopic torque of the pitch angle;

[0038] is the total torque of the roll angle, is the total torque of the pitch angle, is the total torque of the yaw angle.

[0039] Further, in step three, the state equation of the i-th follower UAV attitude system is:

[0040]

[0041] wherein x i,1 = [φ i,1 , φ i,2 , φ i,3 ] T , φ i,1 is the roll angle of the i-th follower UAV, φ i,2 is the pitch angle of the i-th follower UAV, and φ i,3 is the yaw angle of the i-th follower UAV;

[0042] x i,2 = [ω i,x , ω i,y , ω i,z ] T , ω i,x is the roll angular velocity of the i-th follower UAV, ω i,y is the pitch angular velocity of the i-th follower UAV, and ω i,z is the yaw angular velocity of the i-th follower UAV;

[0043] is the total torque of the roll angle of the i-th follower UAV, is the total torque of the pitch angle of the i-th follower UAV, is the total torque of the yaw angle of the i-th follower UAV;

[0044] x i,1 , x i,2 ∈ R 3 is the state vector;

[0045] u i (t) is the device control input, and u i (t) = [u i,1 (t), ui,2 (t),u i,3 (t)] T ∈R 3 ;

[0046] y i is the device control output, and y i ∈R 3 ;

[0047] O i = diag{O i,1 , O i,2 , O i,3} T where

[0048] where,

[0049]

[0050]

[0051]

[0052] where,

[0053]

[0054]

[0055]

[0056] Δ i,m is a bounded time-varying external disturbance, and Δ i,m ∈R 3 (m = 1, 2), assuming that the output x i,1 can be directly measured, while other states x i,2 are not measurable.

[0057] Further, in step four, the design of the dynamic event-triggered mechanism is:

[0058]

[0059]

[0060]

[0061] where j = 1, 2, 3, t is time, t i,h is the input update time;

[0062] ν i(t) is the actual controller, and v i (t) = [v i,1 (t), v i,2 (t), v i,3 (t)] T ∈ R 3 ;

[0063] R i (t) is the difference between the actual controller v i (t) and the device control input u i (t), and R i (t) = [R i,1 (t), R i,2 (t), R i,3 (t)] T , where R i,1 (t) = v i,1 (t) - u i,1 (t), R i,2 (t) = v i,2 (t) - u i,2 (t), and R i,3 (t) = v i,3 (t) - u i,3 (t);

[0064] ξ i (t) ∈ R, and the dynamics of ξ i (t) are designed as

[0065] and where ξ i (0) ∈ (0, 1), φ i > 0, D i > 0; in addition, there exists where and are upper bounds;

[0066] When the triggering condition |R i,j (t)| ≥ ξ i (t)|u i,j (t)| + τ i (t) is satisfied, the device control input will be updated;

[0067] When the system is running normally, the device control input u i (t) = v i (t i,h ), at this time v i (t i,h ) is a constant.

[0068] Further, in step five, the fuzzy logic state observer is:

[0069]

[0070] wherein, is an estimate of the unknown continuous function f i,m , m = 1, 2;

[0071] z i is the observation error, which

[0072] is the system state vector, and is an estimate of ;

[0073] W i,m is the weight matrix, and s i,m is the number of fuzzy rules;

[0074] is an estimate of W i,m ;

[0075] is the membership function vector, and a Gaussian function is selected wherein, and η r represent the center of the receptive field and the width of the Gaussian function, respectively;

[0076] k i,1 , k i,2 > 0.

[0077] Further, in step six, a coordinate transformation is defined first:

[0078]

[0079] wherein, α i,1 is a virtual controller, and y d is the desired Euler angle trajectory of the virtual leader;

[0080] The consensus tracking error of the ith follower UAV system is:

[0081]

[0082] Further, in step seven, the Lyapunov function V i,1 is:

[0083]

[0084] wherein,

[0085] The virtual controller a i,1 and the uncertain parameter adaptive law are respectively:

[0086]

[0087]

[0088] wherein b i = [b i,1 , b i,2 , b i,3 ] T , b i,1 , b i,2 , b i,3 ∈ R,

[0089]

[0090] is a matrix, wherein the matrix elements are the cubic of the matrix elements corresponding to ;

[0091] μ i,1 , η i,1 , p i,1 , q i,1 , Ξ ∈ R + ;

[0092] The Lyapunov function V i,2 is:

[0093]

[0094] wherein,

[0095] The actual controller v i (t) is:

[0096]

[0097] wherein a i,2 is an intermediate control function, and

[0098]

[0099] μ i,2 , η i,2 ∈ R + , Λ i = [Λ i,1 , Λ i,2 , Λ i,3 ]T ,

[0100] and

[0101]

[0102]

[0103] the uncertain parameter adaptive law are respectively:

[0104]

[0105] wherein p i,2 q i,2 ∈R + , is a matrix, wherein the matrix elements are the cube of the matrix elements corresponding to .

[0106] The present application can achieve the beneficial effects:

[0107] 1, in order to maintain the high transmission efficiency of the controller and reduce the communication burden, the improved dynamic event triggering mechanism is put forward, which is the first attempt to deal with multiple input. Compared with the static event triggering mechanism in document 【5】, the dynamic event triggering mechanism has dynamic adjustment threshold, and the event triggering mechanism provides a more flexible way in adjusting data transmission, thereby effectively saving the communication resources. Compared with the dynamic event triggering mechanism given in document 【6】, the dynamic event triggering mechanism of the present application can further reduce the communication burden. The simulation results show that this mechanism greatly reduces the number of triggering, saves the communication resources, and reduces the communication burden.

[0108] 2, the present application uses fuzzy logic system to approximate the unknown function and unmodeled part in the system, and uses adaptive technology to estimate the optimal weight of the fuzzy system online, thereby compensating for the influence on the system.

[0109] 3, the multi-actual controller of unmanned aerial vehicle put forward in the present application fully considers the possible unmeasurable state in reality, so that the controller is more in line with the actual situation. The fixed time state observer based on fuzzy logic system estimates the unmeasurable state, and solves the situation that the unmeasurable state may appear in the system. The designed fuzzy logic state observer has a small state estimation error.

[0110] 4, the multi-actual controller of unmanned aerial vehicle put forward in the present application can realize the bounded stability of the fixed time multi-unmanned aerial vehicle attitude error.

[0111] 5、The multi-unmanned aerial vehicle virtual controller provided by the application uses hyperbolic tangent function to process singular problems that may occur in the derivative of the virtual controller, thereby avoiding chattering phenomenon. Compared with many existing documents that solve singularities by using segmented functions, the application greatly reduces the calculation complexity. BRIEF DESCRIPTION OF DRAWINGS

[0112] Figure 1 is a non-directional communication topology structure of a multi-unmanned aerial vehicle of an embodiment of the application.

[0113] Figure 2 is a curve of an attitude output trajectory and an expected trajectory of three follower unmanned aerial vehicles provided by an embodiment of the application.

[0114] Figure 3 is a consistency tracking error curve of three follower unmanned aerial vehicles provided by an embodiment of the application.

[0115] Figure 4 is a device control input curve of a controller of three follower unmanned aerial vehicles provided by an embodiment of the application.

[0116] Figure 5 is a first coordinate of a state of three follower unmanned aerial vehicles and an observer trajectory curve thereof provided by an embodiment of the application.

[0117] Figure 6 is a second coordinate of a state of three follower unmanned aerial vehicles and an observer trajectory curve thereof provided by an embodiment of the application.

[0118] Figure 7 is a third coordinate of a state of three follower unmanned aerial vehicles and an observer trajectory curve thereof provided by an embodiment of the application.

[0119] Figure 8 is an observer error curve of a first state of three follower unmanned aerial vehicles provided by an embodiment of the application.

[0120] Figure 9 is an observer error curve of a second state of three follower unmanned aerial vehicles provided by an embodiment of the application.

[0121] Figure 10 is a trigger transient and a time interval of three follower unmanned aerial vehicles provided by an embodiment of the application.

[0122] Figure 11 is a trajectory curve of a norm of an uncertain parameter of three follower unmanned aerial vehicles provided by an embodiment of the application. and

[0123] Figure 12 is a trajectory curve of a norm of an uncertain parameter of three follower unmanned aerial vehicles provided by an embodiment of the application. ​And the trajectory curve of the norm of DETAILED DESCRIPTION

[0124] The application will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0125] A multi-UAV fixed-time dynamic event-triggered output feedback control method, in this embodiment, taking a hexacopter UAV as an example, includes the following steps:

[0126] Step one: determine the communication topology structure of multiple hexacopter UAVs.

[0127] The communication relationship between N follower UAVs is described by an undirected graph to express the information exchange relationship, wherein represents the set of N follower UAVs, represents the set of edges, represents the adjacency matrix with non-negative adjacency elements.a i,k represents the degree of dependence of the ith follower UAV on the kth follower UAV. The neighbor set of the ith follower UAV n i is represented as If the ith follower UAV can directly obtain information from the kth follower UAV, i.e., the kth follower UAV a i,k > 0, otherwise, a i,k = 0. For an undirected graph, it is specified that a i,k = a k,i , wherein a k,i represents the degree of dependence of the kth follower UAV on the ith follower UAV. The characteristic of the in-degree matrix is wherein The Laplacian matrix of the undirected graph is represented as The leader adjacency matrix of the undirected graph is defined as wherein a i,0 represents the degree of dependence of the ith follower UAV on the leader. If the ith UAV can directly obtain information from the leader, then a i,0 > 0, otherwise, a i,0 = 0.

[0128] Step two: model the attitude physical characteristics of a single hexacopter UAV to obtain a single hexacopter UAV attitude system model.

[0129] Taking the center of gravity of the hexacopter UAV as the origin, the earth-fixed inertial system and the body-fixed inertial system are considered, wherein Oe is the origin of the earth-fixed inertial coordinate system, x e is the x-axis of the earth-fixed inertial coordinate system, y e is the y-axis of the earth-fixed inertial coordinate system, z e is the z-axis of the earth-fixed inertial coordinate system, O b is the origin of the body-fixed inertial coordinate system, x b is the x-axis of the body-fixed inertial coordinate system, y b is the y-axis of the body-fixed inertial coordinate system, z b is the z-axis of the body-fixed inertial coordinate system. In the coordinate system , φ = [φ1, φ2, φ3] T represents the Euler angles of the hexacopter, where φ1is the roll angle of the hexacopter, φ2is the pitch angle of the hexacopter, and φ3is the yaw angle of the hexacopter. In the coordinate system , x = [x, y, z] is the coordinate of the center of gravity of the hexacopter, where x represents the x-axis coordinate, y represents the y-axis coordinate, and z represents the z-axis coordinate. In the coordinate system , ω = [ω x , ω y , ω z ] T represents the angular velocity of the center of gravity of the hexacopter, where ω x is the roll angular velocity of the hexacopter, ω y is the pitch angular velocity of the hexacopter, and ω z is the yaw angular velocity of the hexacopter.

[0130] According to the Newton-Euler equation, the single hexacopter attitude system model is as follows:

[0131]

[0132] wherein is the transformation matrix, is the inertia matrix. and represent the gyroscopic torque and the total torque, respectively, wherein is the gyroscopic torque of the roll angle, is the gyroscopic torque of the pitch angle, is the total torque of the roll angle, is the total torque of the pitch angle, is the total torque of the yaw angle. The transformation matrix is as follows:

[0133] The attitude system model (1) is converted into the following form:

[0134] and

[0135] Step three: Obtain the state equation of the i-th follower UAV attitude system according to the single six-rotor UAV attitude system model.

[0136] Define x i,1 = [φ i,1 , φ i,2 , φ i,3 ] T , x i,2 = [ω i,x , ω i,y , ω i,z ] T and Here, φ i,1 represents the roll angle of the i-th follower UAV, φ i,2 represents the pitch angle of the i-th follower UAV, φ i,3 represents the yaw angle of the i-th follower UAV; ω i,x represents the roll angular velocity of the i-th follower UAV, ω i,y represents the pitch angular velocity of the i-th follower UAV, ω i,z represents the yaw angular velocity of the i-th follower UAV; represents the total torque of the roll angle of the i-th follower UAV, represents the total torque of the pitch angle of the i-th follower UAV, represents the total torque of the yaw angle of the i-th follower UAV.

[0137] Consider a multi-UAV consisting of N UAVs with uncertain dynamics and external disturbances. The state equation of the i-th follower UAV attitude system is represented as:

[0138]

[0139] where x i,1 , x i,2 ∈ R 3 is the state vector, u i (t) = [u i,1 (t), u i,2 (t), u i,3 (t)] T ∈ R 3 and y i ∈ R 3 are the device control input and output, respectively. O i = diag{O i,1 , O i,2 , O i,3} T where and

[0140] wherein

[0141]

[0142] wherein

[0143]

[0144] Δ i,m ∈R 3 (m = 1, 2) are bounded time-varying external disturbances. For each follower UAV system (i = 1,..., N), it is assumed that the output x i,1 can be measured directly, while other states x i,2 are unmeasurable.

[0145] Step four: design a dynamic event-triggered mechanism for the ith follower UAV;

[0146] To maintain the high transmission efficiency of the controller and alleviate the communication burden among system components, an improved adjustable parameter dynamic event-triggered mechanism is introduced for (i = 1,..., N; j = 1, 2, 3)

[0147]

[0148] where t denotes time, v i (t) = [v i,1 (t), v i,2 (t), v i,3 (t)] T ∈R 3 is the actual controller, t i,h is the input update time. R i (t) = [R i,1 (t), R i,2 (t), R i,3 (t)] T is the difference between the actual controller v i (t) and the device control input u i (t), where R i,1 (t) = v i,1 (t) - u i,1 (t), R i,2 (t) = v i,2 (t) - u i,2 (t), R i,3 (t) = v i,3 (t) - u i,3 (t). ξ iThe dynamics of (t) (i = 1,..., N) ∈ R are designed as and where ξ i (0) ∈ (0, 1), φ i > 0, and D i > 0. Moreover, there exist and where and are upper bounds. If the triggering condition |R i,j (t)| ≥ ξ i (t)|u i,j (t)| + τ i (t) is satisfied, the device control input will be updated. When the system is running normally, the device control input u i (t) = v i (t i,h ), where v i (t i,h ) is a constant.

[0149] Remark 1: is a continuous smooth function in the real number field, whose value range is 0 to 1.

[0150] According to the dynamic event-triggered mechanism (5), the relationship between v i,j (t) and u i,j (t) can be easily established as follows:

[0151]

[0152] where ρ 1,j (t) and ρ 2,j (t) (j = 1, 2, 3) are time-varying parameters satisfying |ρ 1,j (t)| ≤ 1 and |ρ 2,j (t)| ≤ 1.

[0153] Remark 2: The main idea of the proposed dynamic event-triggered mechanism is to dynamically adjust the threshold based on the functions ξ i (t) and τ i (t). In addition, compared with the static event-triggered method in 【5】 and the dynamic event-triggered mechanism given in 【6】, the improved dynamic event-triggered mechanism in this paper can further reduce the communication burden.

[0154] Assumption 1: The external disturbance is Δ i,j (i = 1,..., N, m = 1, 2) is bounded, that is, there exist some constants such that

[0155] Assumption 2: Desired Euler angle trajectory y of virtual leader d and its derivative are continuous and bounded functions, i.e., there exists θ∈R ＞0 such that

[0156] Definition 1: Fixed-time stability:

[0157] Consider a dynamical system:

[0158]

[0159] where x∈R n is the state and f(·) represents a nonlinear smooth function. Based on Lyapunov functions, it is assumed that the origin of the system (7) is stable. If for all time there exists a finite convergence time such that x(t) = 0, then the system (7) is finite-time stable. According to the above assumption, if the stability time function has an upper bound, then the system (7) is considered to be fixed-time stable.

[0160] Lemma 1: For the system (7), if a smooth positive definite and radially unbounded function V(x): R n → R satisfies

[0161]

[0162] where 0 < α < 1, β > 1 and η, μ and C are positive, then the origin x = 0 of (7) is actually fixed-time stable. Furthermore, the residual set of the system solution can be written as where settling time is bounded and can be represented as

[0163]

[0164] Scaling inequality:

[0165] Lemma 2: For 1,2 ∈ R, and the following inequality holds

[0166]

[0167] Lemma 3: For a constant Ξ∈R ＞0 and τ∈R, it can be obtained that

[0168]

[0169] Lemma 4: For the variable ζi , assuming unknown constants such that when h i > 1, the following inequalities hold

[0170]

[0171] Lemma 5: For i1> 0, i2> 0, i3> 0, ψ1≥ 0, ψ2≥ 0 and ψ3≥ 0, the following inequalities hold

[0172]

[0173] Lemma 6: For any x1, x2,..., x n ≥ 0 and one can obtain

[0174]

[0175] Lemma 7: For any x1, x2,..., x n ≥ 0 and one can obtain

[0176]

[0177] Lemma 8: In the design of the controller, the fuzzy logic system is used to approximate the unknown continuous function f(x): R n → R c , which is a smooth function on a compact set , for any constant ε > 0 and can be approximated by a fuzzy logic system , i.e., for any constant ε > 0, there exists a fuzzy logic system such that

[0178]

[0179] where W ∈ R s×c is the weight matrix and s is the number of fuzzy rules. is the membership function vector and is chosen as the following Gaussian function: where v r = [v r,1 ,..., v r,n ] T and η r represent the center of the receptive field and the width of the Gaussian function, respectively.

[0180] Remark 3: According to the definitions of and , we have and hold. The characteristics of this feature will be used to separate state variables in the following analysis.

[0181] Step 5: For the i-th follower drone, establish a fuzzy logic state observer to estimate the unmeasurable state of the system and obtain the estimated value of the system state;

[0182] According to Lemma 8, assume an unknown smooth function. and It can be approximated by a fuzzy logic system and satisfies

[0183]

[0184] in It is a state vector. yes The estimate, It is the weight matrix, s i,m It represents the number of fuzzy rules. It is a membership function vector and Choose the following Gaussian function: in and η r represents the center of the receptive field and the width of the Gaussian function, respectively.

[0185] Define the approximation error ε respectively i,m ∈R 3 and δ i,m ∈R 3 for

[0186]

[0187] in W i,m The estimate.

[0188] Assumption 3: Error ε i,m and δ i,m It is bounded, meaning that there exist some positive constants. and Make and

[0189] The observation error is defined as in It is the system state vector The estimate.

[0190] To estimate the unmeasurable states in system (2), the following fuzzy state observer was constructed:

[0191]

[0192] in is an unknown continuous function f i,m (m = 1, 2) of the estimate k i,1 ,k i,2 > 0.

[0193] For the sake of the following analysis, the system (2) can be rewritten as

[0194]

[0195] where κ i = [k i,1 ,k i,2 ] T , Γ i,1 = [1, 0] T , Γ i,2 = [0, 1] T , Φ i = [0, 1] T , C i = [1, 0] T , I3 denotes the third-order identity matrix and represents the Kronecker product. The vector κ i is chosen such that K i is a Hurwitz matrix, i.e., for any matrix P i = P i T > 0, there exists a symmetric matrix such that

[0196]

[0197] From (19), the fuzzy state observer can be summarized as follows (i = 1, 2, …, N):

[0198]

[0199] Then, the following observer error z i (i = 1, 2, …, N) can be obtained from (20) and (22):

[0200]

[0201] where

[0202] Step six: construct the consensus tracking error of the i-th follower UAV system according to the estimated value of the system state and the desired Euler angle trajectory of the virtual leader.

[0203] Define the following coordinate transformation:

[0204]

[0205] where the virtual controller a i,1 will be defined later, y d is the desired Euler angle trajectory of the virtual leader.

[0206] Define the consensus tracking error of the ith follower UAV system:

[0207]

[0208] To analyze the stability of the consensus tracking error, construct a Lyapunov function V L based on the consensus tracking error as follows:

[0209]

[0210] where and

[0211] Remark 4: Define Thus, we have Therefore, the Lyapunov function (26) can be rewritten as and

[0212] Remark 5: From [1], we have, is a positive definite matrix, so its eigenvalues λ1,..., λ N are non-negative. According to the matrix theory, σ 1,1 , σ 1,2 , σ 1,3 ,..., σ N,1 , σ N,2 , σ N,3 can be chosen as an orthonormal basis of R 3N , where σ 1,1 , σ 1,2 , σ 1,3 ,..., σ N,1 , σ N,2 , σ N,3 are the eigenvectors corresponding to the eigenvalues λ1,..., λ N of the matrix . Define M = (σ 1,1 ,..., σ N,3 ) ∈ R 3N×3N , then we have M T M = MM T = I 3N .

[0213] Then, the Lyapunov function (26) can also be written as:

[0214]

[0215] where H = diag (λ1I3,..., λ N I3) and Π = MH -1 M T .

[0216] For (27), it can be easily obtained that

[0217]

[0218] where λ min (Π) and λ max (Π) denote the minimum and maximum eigenvalues of matrix Π, respectively.

[0219] To analyze the stability of the observer error, a Lyapunov function V i,0 is constructed based on the observer error as follows:

[0220]

[0221] Further, the derivative of V i,0 is

[0222]

[0223] By using Young’s inequality and Cauchy’s inequality, the following inequality holds:

[0224]

[0225] where denotes the F-norm of a matrix.

[0226] Substituting (31) into (30) gives

[0227]

[0228] Remark 6: According to (28), it can be obtained that

[0229]

[0230] Thus, there exists

[0231] where

[0232]

[0233] Therefore, z i is bounded, in other words, there exists a constant such that

[0234] Step 7: Based on the consistency tracking error and dynamic event triggering mechanism, and within the framework of the backstepping technique, first construct the Lyapunov function V. i,1 According to the Lyapunov function V i,1 Building a virtual controller α i,1 Adaptive Law of Uncertain Parameters Reconstruct the Lyapunov function V i,2 According to the Lyapunov function V i,2 Building an actual controller ν i (t) and the adaptive law of uncertain parameters

[0235] Within the framework of the backstepping technique, it is necessary to design a virtual controller in the first step and obtain the actual controller in the last step.

[0236] Step 1: Construct the Lyapunov function V i,1 as follows:

[0237]

[0238] in

[0239] According to (2), (18) and (24), e i,1 The time derivative is

[0240]

[0241] Furthermore, V i,1 The derivative is

[0242]

[0243] Using the properties of Young's inequality, Cauchy's inequality, and fuzzy basis functions It can be obtained

[0244] Furthermore, substituting (38) into (37) yields

[0245]

[0246] Next, construct the virtual controller α. i,1 Adaptive Law of Uncertain Parameters for

[0247]

[0248]

[0249] Where b i =[b i,1 ,b i,2 ,bi,3 ] T , and is a matrix, where the matrix elements are the cube of the matrix elements of b i,1 ,b i,2 ,b i,3 ∈R and μ i,1 ,η i,1 ,p i,1 ,q i,1 ,Ξ∈R + .

[0250] Substituting (40) and (41) into (39), the following equation holds

[0251]

[0252] Since μ i,1 ,η i,1 ,p i,1 > 0, by using Lemma 7 and Young's inequality, we have

[0253]

[0254] and

[0255]

[0256] Since and we can define them as and where and the following relationship holds

[0257] Thus, we can easily obtain

[0258]

[0259] Since By using Lemma 2, we have

[0260]

[0261] where γ i,1 > 0.

[0262] Thus, according to Lemma 7, we have

[0263]

[0264] Substituting (43), (44) and (47) into (42) and using Lemma 3, we have​

[0265]

[0266] where

[0267]

[0268] Second step: Construct Lyapunov function V i,2 As follows:

[0269]

[0270] where

[0271] Based on (19) and coordinate transformation e i,2 The derivative of

[0272]

[0273] where and

[0274] According to Young inequality, it is easy to obtain

[0275]

[0276] Thus, The following form can be derived

[0277]

[0278] In order to ensure the high transmission efficiency of the controller, an improved dynamic event-triggered mechanism is proposed in this paper.

[0279] Based on the dynamic event-triggered mechanism as follows:

[0280]

[0281] The device control input u i (t) is designed as follows:

[0282] When t∈[t i,h ,t i,h+1 ), the system is running normally, the device control input u i (t) = v i (t i,h ), at this time, u i (t) is a constant; if the trigger condition is met, the device control input u i (t) = v i (t).

[0283] Design the actual controller vi (t) and the intermediate control function a i,2 ∈ R 3 is

[0284]

[0285] where μ i,2 ,η i,2 ∈ R + ,Λ i = [Λ i,1 ,Λ i,2 ,Λ i,3 ] T , and and Λ i,1 ,Λ i,2 ,Λ i,3 , In addition, define where Therefore, the expressions of F i,2 and L i,2 are as follows

[0286]

[0287] Then, by combining v i,j (t) (j = 1, 2, 3) and (6), we can get

[0288] Therefore, it is easy to get

[0289]

[0290] By substituting (55) and (58) into (53) and combining Lemma 3, we can get

[0291]

[0292] Design of adaptive law for uncertain parameters is

[0293]

[0294] where p i,2 ,q i,2 ∈ R + . And is a matrix, where the matrix elements are the cube of the matrix elements corresponding to .

[0295] In addition, substitute (60) into (59) and refer to the processing method of , we get

[0296]

[0297] where γ i,m > 0 (m = 1, 2), and

[0298]

[0299] Step eight: Based on the actual controller v i (t) controls all the follower UAVs to track the desired Euler angle trajectory of the virtual leader.

[0300] The above control reasoning and analysis can be summarized as the following theorem.

[0301] Theorem 1: For multi-UAVs (2) with undirected fixed graph under assumptions 1-3, the control scheme including the virtual controller (40) and the actual controller (55) as well as the uncertain parameter adaptive laws (41) and (60) can ensure that all the signals in the closed-loop system are bounded and the tracking error converges to a small region around zero within a fixed time Tf. is expressed as where In addition, there is a lower bound for the time interval between event triggers, which means that Zeno phenomenon does not occur.

[0302] Proof: The Lyapunov function of the whole system is constructed as

[0303]

[0304] Referring to (28), the following inequality

[0305]

[0306] where denotes the minimum eigenvalue of a matrix and denotes the maximum eigenvalue of a matrix.

[0307] According to (61) and (64), can be derived as

[0308]

[0309] From Remark 6, it can be seen that ‖z i ‖ is bounded. Since and using Lemma 4, it can be obtained that

[0310]

[0311] where

[0312] Using Lemma 5, we have

[0313]

[0314] Thus, we have

[0315]

[0316] According to (64) and substituting (68) into (65), we have

[0317] where

[0318] Define

[0319] According to (28), Lemma 6 and Lemma 7, we have

[0320]

[0321] where and

[0322] Define

[0323]

[0324] where is achieved by choosing appropriate parameters.

[0325] Define again and

[0326] Substituting (70) into (69) and using Lemma 6 and Lemma 7, can be rewritten as

[0327]

[0328] Define η = min{η1, η2} and

[0329] Moreover, using Lemma 6 and Lemma 7, we have

[0330]

[0331] where

[0332] Based on (74) and Lemma 1, it is not difficult to obtain Therefore, V is bounded and by the boundedness of V we know that ||b i ||, ||e i,m ||, ||z i || and is bounded.||z i,m ||According to||z i The boundedness of||a i,m ||and are bounded, their boundedness is obtained by||b i ||,||e i,m ||,||z i,m and The boundedness of e i,1 = x i,1 -y d and x i,1 ||and are bounded. And it can be inferred from that is bounded. The device control input||u i (t)||and the actual controller|ν i (t)||are bounded by (6) and (55). Therefore, all signals of the closed-loop system are bounded, and the proof is complete. And by using Lemma 1, according to (73), the settling time is:

[0333] In addition, according to R i,j (t) = v i,j (t) - u i,j (t) (j = 1, 2, 3) get Because for any u i,j (t) is a constant, so it can be easily obtained According to (55) and (60), v i,j (t) is differentiable and the derivative of v i,j (t) is bounded, that is, there exists such that By and , The lower bound of must satisfy

[0334] Therefore, The lower bound of t * , satisfies That is, the Zeno phenomenon is avoided.

[0335] Matlab simulation results

[0336] To further verify the effectiveness of the designed control method, a multi-agent system consisting of three follower UAVs with external disturbances and a virtual leader was simulated. The communication network topology among the four hexacopter UAV systems is shown in Fig. 6 (N = 3). Figure 1

[0337] The desired Euler angle trajectory of the virtual leader was set as y d = [y d,1 ,y d,2 ,y d,3 ] T , where y d,1 = 0.35cos(0.7t - 1.8) deg is the trajectory of the desired roll angle of the virtual leader, y d,2 = -0.35sin(0.7t - 1.8) deg is the trajectory of the desired pitch angle of the virtual leader, and y d,3 = 0.35cos(0.7t - 1.8) - 0.21sin(0.7t - 1.8) deg is the trajectory of the desired yaw angle of the virtual leader. In this case, k and k were chosen to simulate the external disturbances in the UAV flight environment, where

[0338]

[0339] The connection weight matrix between the follower UAVs and the virtual leader was The adjacency matrix A was chosen as follows:

[0340]

[0341] In addition, the virtual control a i,1 and the actual controller v i (t) can be represented as: (i = 1, 2, 3)

[0342]

[0343] and the adaptive law was chosen as follows: (i = 1, 2, 3)

[0344]

[0345] and the dynamic event-triggered mechanism

[0346]

[0347] In the simulation, the initial state can be chosen as x i,1 (0) = [-0.5, 0, -0.4] T , x i,2 ​(0) = [-1, -1.5, -0.5] T , and where [·] 15×3 denotes a matrix with 15x3 dimensions. The simulation parameters are shown in Table 1 (i = 1, 2, 3; m = 1, 2; q = 2, 3).

[0348] Table 1: Simulation parameter table

[0349]

[0350] The simulation results are shown in Figures 2-12 .

[0351] Figure 2 The output tracking trajectories of the three UAVs in three-dimensional space are disclosed, reflecting the consistent tracking performance of the multi-UAV system. Figure 3 As can be seen in the figure, the consistency tracking error can converge to a small area near the zero point in the figure, and it is not difficult to see that the required tracking capability has been achieved. Figure 4 The device control input trajectories of the three follower UAVs are shown. Figures 5-7 The trajectories of the states of the three follower UAVs and their observers are shown. In Figure 8 and Figure 9 , it can be easily seen that the fuzzy logic system state observer can accurately estimate the unobservable states of the multi-UAV system. Figure 10 By observing the horizontal and vertical coordinates, all the triggering transients and triggering intervals can be seen. In addition, as can be seen in Figure 10 , the number of triggers is limited within 40 seconds. That is, within a certain period of time, the control signal will not be transmitted infinitely, which means that the Zeno phenomenon will not occur. The norms of the uncertain parameters and are shown in Figures 11-12 . As can be seen in Figures 11-12 , the norms of the uncertain parameters and are bounded. The number of triggers of the multi-UAV system under three different triggering mechanisms (the method of the present embodiment, the static event-triggered mechanism 【5】, and the dynamic event-triggered mechanism 【6】 ) is obtained by Matlab, as shown in Table 2. It can be seen that compared with the traditional static event-triggered mechanism 【5】 and the dynamic event-triggered mechanism 【6】, the proposed dynamic event-triggered mechanism significantly reduces the number of triggers, thereby reducing the communication frequency and alleviating the communication burden.

[0352] Table 2: Trigger number table under different triggering mechanisms

[0353] This embodiment Static event triggering mechanism 【5】 Dynamic event triggering mechanism 【6】 Drone 1 889 938 963 Drone 2 781 852 801 Drone 3 465 533 501

Claims

1. A multi-UAV fixed-time dynamic event-triggered output feedback control method, characterized in that: The method comprises the following steps: Step 1: determining the communication topology of a plurality of hexacopter unmanned aerial vehicles (UAVs); Step 2: modeling the attitude physical characteristics of a single hexacopter UAV to obtain a single hexacopter UAV attitude system model; Step 3: obtaining an i-th follower UAV attitude system state equation according to the single hexacopter UAV attitude system model; Step 4: designing a dynamic event triggering mechanism for the i-th follower UAV; Step 5: establishing a fuzzy logic state observer for the i-th follower UAV to estimate the unmeasurable state of the system and obtain an estimated value of the state of the system; Step 6: constructing a consensus tracking error of the i-th follower UAV system according to the estimated value of the state of the system and a desired Euler angle trajectory of a virtual leader. Step 7: Based on the consistency tracking error and dynamic event triggering mechanism, under the backstepping technique framework, first construct the first Lyapunov function V. i,1 According to the first Lyapunov function V i,1 Building a virtual controller α i,1 Adaptive law for the first uncertain parameter Construct the second Lyapunov function V i,2 According to the second Lyapunov function V i,2 Building an actual controller ν i (t) and the adaptive law of the second uncertain parameter Step eight: Based on the actual controller v i (t) Control all follower drones to track the desired Euler angle trajectory of the virtual leader.

2. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 1, wherein: In step one, the communication topology of the plurality of hexacopter UAVs is a communication relationship between N follower UAVs by an undirected graph to express the information exchange relationship, wherein, represents a set of N follower UAVs, represents a set of edges, represents an adjacency matrix with non-negative adjacency elements; a i,k represents the degree of dependence of the i-th follower UAV on the k-th follower UAV, i = 1, 2, …, N; the neighbor set of the i-th follower UAV n i is represented as If the i-th follower UAV can directly obtain information from the k-th follower UAV, i.e., the k-th follower UAV a i,k > 0, otherwise a i,k = 0; for the undirected graph, a i,k = a k,i , a k,i represents the degree of dependence of the k-th follower UAV on the i-th follower UAV; the characteristic of the in-degree matrix is wherein the Laplacian matrix of the undirected graph is represented as The leader adjacency matrix of the undirected graph is defined as wherein a i,0 represents the degree of dependence of the i-th follower UAV on the leader, if the i-th UAV can directly obtain information from the leader, then a i,0 > 0, otherwise a i,0 = 0.

3. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 2, wherein: In Step 2, the single hexacopter UAV attitude system model is: wherein is a transformation matrix, is an inertia matrix, is a gyroscopic torque, is a total torque; φ = [φ1, φ2, φ3] T φ1is a roll angle, φ2is a pitch angle, and φ3is a yaw angle. ω = [ω x , ω y , ω z ] T , ω x is the roll angular velocity, ω y is the pitch angular velocity, and ω z is the yaw angular velocity; is the gyroscopic torque of the roll angle, is the gyroscopic torque of the pitch angle; is the total torque of the roll angle, is the total torque of the pitch angle, is the total torque of the yaw angle.

4. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 3, wherein: In Step 3, the i-th follower UAV attitude system state equation is: wherein x i,1 = [φ i,1 , φ i,2 , φ i,3 ] T , φ i,1 is the roll angle of the i-th follower drone, φ i,2 is the pitch angle of the i-th follower drone, and φ i,3 is the yaw angle of the i-th follower drone. x i,2 = [ω i,x , ω i,y , ω i,z ] T , ω i,x is the roll angular velocity of the i-th follower drone, ω i,y is the pitch angular velocity of the i-th follower drone, and ω i,z is the yaw angular velocity of the i-th follower drone. is the total torque of the roll angle of the i-th follower drone, is the total torque of the pitch angle of the i-th follower drone, is the total torque of the yaw angle of the i-th follower drone; x i,1 ,x i,2 ∈R 3 is the state vector; u i (t) is a device control input, and u i (t) = [u i,1 (t), u i,2 (t), u i,3 (t)] T ∈ R 3 ; y i is a device control output, and y i ∈R 3 ; O i = diag{O i,1 ,O i,2 ,O i,3} T where wherein, wherein, Δ i,m is a bounded time-varying external disturbance, and Δ i,m ∈ R 3 (m = 1, 2), assuming that the outputs x i,1 can be directly measured, while other states x i,2 are not.

5. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 4, wherein: In Step 4, the dynamic event triggering mechanism is designed as: where j = 1, 2, 3, t is time, t i,h is the input update time; ν i (t) is the actual controller, and ν i (t) = [ν i,1 (t), ν i,2 (t), ν i,3 (t)] T ∈ R 3 ; R i (t) is the actual controller ν i (t) and device control input u i The difference between (t) and R i (t)=[R i,1 (t),R i,2 (t),R i,3 (t)] T , where R i,1 (t)=ν i,1 (t)-u i,1 (t), R i,2 (t)=ν i,2 (t)-u i,2 (t), R i,3 (t)=ν i,3 (t)-u i,3 (t); ξ i (t) ∈ R, and ξ i The dynamics of ξ And where ξ i (0)∈(0,1), φ i >0, D i >0; moreover, there exists where and is an upper bound; When the triggering condition |R i,j (t)|≥ξ i (t)|u i,j (t)|+τ i (t) is satisfied, the device control input will be updated; When the system is operating normally, the plant control input u i (t) = v i (t i,h ), where v i (t i,h ) is a constant.

6. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 5, wherein: In Step 5, the fuzzy logic state observer is: wherein is an estimate of the unknown continuous function f i,m m = 1,2; z i is the observation error, which is the system state vector, and is the estimate of . W i,m is a weight matrix, and s i,m is the number of fuzzy rules; is W i,m estimate; is a membership function vector, and Gaussian function is selected wherein, and η r respectively represent the center of the receptive field and the width of the Gaussian function; k i,1 ,k i,2 > 0.

7. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 6, wherein: In Step 6, a coordinate transformation is defined as: where a i,1 is the virtual controller, y d is the desired Euler angle trajectory of the virtual leader; The consensus tracking error of the i-th follower UAV system is:

8. The multi-UAV fixed-time dynamic event-triggered output feedback control method of claim 7, wherein: In step seven, the Lyapunov function V i,1 is: wherein I3denotes a third order identity matrix, denotes a Kronecker product; symmetric matrix The virtual controller a i,1 And uncertain parameter adaptive law Respectively: where b i = [b i,1 ,b i,2 ,b i,3 ] T , b i,1 ,b i,2 ,b i,3 ∈ R, is a matrix, where the matrix elements are the cubes of the matrix elements of is a matrix, where the matrix elements are the cubes of the matrix elements of μ i,1 ,η i,1 ,p i,1 ,q i,1 ,Ξ∈R + ; The Lyapunov function V i,2 is: wherein The actual controller v i (t) is: where α i,2 is an intermediate control function, and μ i,2 ,η i,2 ∈R + ,Λ i =[Λ i,1 ,Λ i,2 ,Λ i,3 ] T , and The uncertain parameter adaptive law Respectively: where p i,2 , q i,2 ∈ R + , is a matrix whose matrix elements are the cube of the matrix elements of .

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