Optimal rendezvous of multiple uavs in weighted imbalanced directed communication networks

Abstract

The optimal gathering algorithm of multiple unmanned aerial vehicles in a weight unbalanced directed communication network comprises the following steps: when unmanned aerial vehicles gather in a same task area, a local target function of each unmanned aerial vehicle is determined, and a sum of all local target functions is set as a total target function of the whole unmanned aerial vehicle cluster; in order to make the unmanned aerial vehicle cluster meet the condition that the total target function is minimum and the communication network graph is a weight unbalanced directed graph, an optimal gathering is reached, and a balance point of the unmanned aerial vehicle gathering is obtained through the distributed optimization control method; it is proved in a theoretical level that the balance point obtained by the distributed optimization control algorithm is the optimal gathering point meeting the condition that the total target function is minimum; it is proved that each unmanned aerial vehicle can gather to the optimal gathering point according to the algorithm; the communication network of the unmanned aerial vehicle cluster is set as a weight unbalanced directed graph which is more close to a real task condition, and the important factor of gathering time is considered, so that the unmanned aerial vehicle can automatically search the optimal gathering point within a given time.

Description

Technical Field

[0001] This invention belongs to the field of unmanned aerial vehicle (UAV) swarm control technology, and particularly relates to the optimal swarming algorithm for multiple UAVs in a weight-unbalanced directed communication network. Background Technology

[0002] In recent years, drone technology has become increasingly sophisticated. With its advantages such as unmanned operation, higher safety, and insensitivity to terrain, drones have gradually entered various industries and are used to complete various challenging tasks. However, a single drone often cannot meet the increasingly complex task environment of today. Therefore, the swarm control of multiple drones is a key research issue in the current drone field, and the swarming of drones is a crucial link in drone swarm control.

[0003] Multi-UAV swarm control methods are broadly classified into two types: centralized control and distributed control. Centralized control requires a central control center to manage the entire UAV swarm. Due to the limited computational capacity of the control center, large-scale control is difficult, and the cost is high. If the control center is destroyed, the UAV swarm becomes paralyzed and unable to continue operating. In contrast, distributed control does not rely on commands from a control center. It achieves swarm control through information exchange among UAVs within the swarm. Even if a UAV malfunctions, it does not affect the overall swarm mission. Furthermore, swarm control offers greater flexibility, scalability, and robustness. Currently, research on distributed control algorithms for multi-agent systems like multi-UAVs and UAV swarming problems has been conducted both domestically and internationally. The relevant literature includes Nedic A, Ozdaglar A. Distributed Subgradient Methods for Multi-Agent Optimization[J].IEEE Transactions on Automatic The distributed subgradient algorithm proposed in Control, 2009, 54(1):48-61, enables the state variables of all multi-agents to converge to an optimal solution. The distributed alternating direction multiplier method proposed in the literature MOTA JF, XAVIER JM, AGUIARP M, et al. D-ADMM: A communication-efficient distributed algorithm for separable optimization[J]. IEEE Transactions on Signal Processing, 2013, 61(10):2718–2723, optimizes the convergence rate of distributed optimization.References: Xu W, Yang S. Projection-based Dynamics for Distributed Optimization Subject to General Constraints [C] / / Proceedings of the 37th Chinese Control Conference (B). Wuhan: Technical Committee on Control Theory, Chinese Association of Automation, 2018: 766-770. Further research on distributed optimization problems with general constraints under undirected graphs; Chen G, Yang Q, Song Y, et al. A Distributed Continuous-Time Algorithm for Nonsmooth Constrained Optimization [J]. IEEE Transactions on Automatic Control, 2010, 65(11): 4914–4921. The distributed convex optimization problem of nonsmooth local objective functions under dual constraints of layout inequality constraints and coupling equality constraints is studied; Li L, Yu Y, Li X, et al. Exponential convergence of distributed optimization for heterogeneous linear multi-agent systems over Unbalanced digraphs[J].Automatica,2022,141:110259-110269. Proportional-integral control technology is adopted to achieve the optimal distributed distribution of heterogeneous linear multi-agent systems; NEDIC A, OLSHEVSKY A. Distributed optimization over time-varying directed-graphs[J].IEEE Transactions on Automatic Control,2015,60(3):601–615. Combining the Push-sum protocol and the distributed subgradient algorithm, an improved Subgradient-push algorithm is proposed, which realizes the distributed optimization of directed switching networks.References XU J, ZHU S, SOH YC, et al. Augmented distributed gradient methods for multi-agent optimization under uncoordinated constant stepsizes[C] / / 2015 54th IEEE Conference on Decision and Control (CDC). Osaka, Japan: IEEE, 2015: 2055–2060. This paper proposes the original algorithm Aug-DGM based on directed graphs, utilizing dynamic average consistency. References Yang Q, Chen G, Ren J. Continuous-Time Algorithm For Distributed Constrained Optimization Over Directed Graphs[C] / / 2019 IEEE 15th International Conference on Control and Automation (ICCA). Edinburgh, UK: IEEE, 2019: 1020-1025. These researchers, based on weighted directed graphs, solve the distributed optimization problem for non-smooth and non-quadratic objective functions.

[0004] The above literature is based on the fact that the communication network between multiple agents is an undirected graph or a weighted directed network. This is obviously an ideal communication situation for the multi-UAV swarming problem. When multiple UAVs swarm, the communication between them is often interfered with to a certain extent due to the limitations of the UAVs themselves and external conditions. Therefore, the research on the UAV swarming problem based on unbalanced directed graphs is the key.

[0005] Regarding UAV swarming, there is currently limited research both domestically and internationally. Relevant literature includes: Yang Zhengquan, Yang Xiuwei, Chen Zengqiang. Design of Constrained Continuous-Time Distributed Optimization Algorithm in Unbalanced Directed Networks [J / OL]. Guangzhou: Control Theory & Applications. 2022(2022-9-21)[2023-1-23]. http: / / kns.cnki.net / kcms / detail / 44.1240.TP.20220920.1746.044.html Yang ZQ, Yang XW, Cheng ZQ. Continuous time with constraints in general directed networks distributed optimization algorithm design [J / OL]. Guangzhou: Control Theory & Applications. 2022(2022-9-21)[2023-1-23]. http: / / kns.cnki.net / kcms / detail / 44.1240.TP.20220920.1746.044.html Theory & Applications.2022(2022-9-21)[2023-1-23].http: / / kns.cnki.net / kcms / detail / 44.1240.TP.20220920.1746.044.html For the UAV swarming problem, a distributed optimal consensus (CGOC) algorithm based on cooperative game theory is proposed. This method reduces the transmission of redundant information and reduces the impact of finite networks on the task. The literature Zhu Y, Ren W, Yu W, et al. Distributed resource allocation over directed graphs via continuous-time algorithms[J].IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, 51(2):1097-1106. A swarming method for fixed-wing UAVs is proposed. The paper focuses on the speed and trajectory adjustments of fixed-wing UAVs during flight, introducing a time-varying vector field to adjust the flight trajectory of the UAVs according to their flight time, so that multiple UAVs can reach the agreed position simultaneously. Reference

[14] studied the problem of UAVs used for aerial surveillance and mapping in remote areas assembling with dynamic UGVs (UAV refueling vehicles) on the ground when they need to replenish energy. Reference Chen Z, Ma J, Wang X, et al. IEEE 14th International Conference on Control and Automation. Anchorage, AK, USA: IEEE, 2018: 1-6. It proposed a distributed solution strategy based on coordination variables and coordination functions to solve the multi-UAV assembly problem.

[0006] The aforementioned literature has proposed a series of algorithms for the swarming problem of multiple UAVs under various mission scenarios. However, when considering the swarming problem, it lacks consideration of swarming time or treats swarming time as only a part of the algorithm optimization result, which has no practical value. Therefore, this application studies an optimal swarming algorithm for multiple UAVs based on the above research. It sets the communication network of the UAV swarm as a weighted unbalanced directed graph that is closer to the real mission situation, and considers the important factor of swarming time, so that the UAVs can automatically search for the optimal swarming point within a given time to achieve the optimal swarming effect. Summary of the Invention

[0007] The technical problem solved by this invention is achieved through the following technical solution:

[0008] The optimal aggregation algorithm for multiple UAVs in a weight-imbalanced directed communication network includes the following steps:

[0009] 1) When N drones gather in the same mission area, determine the local objective function of each drone and set the sum of all local objective functions as the total objective function of the entire drone swarm;

[0010] 2) To enable the UAV swarm to achieve optimal aggregation under the conditions of minimizing the overall objective function and the communication network graph being a weighted unbalanced directed graph, a corresponding distributed optimization control method is designed, and the equilibrium point of controlling the UAV swarm is obtained through this distributed optimization control method.

[0011] 3) Optimality analysis: Theoretically prove that the equilibrium point obtained by this distributed optimization control method is the required optimal cluster point;

[0012] 4) Convergence analysis: Prove that each drone can be controlled by the algorithm to gather at the optimal gathering point.

[0013] In step 1), when the drones assemble, the assembly point of the N drones is required to be globally optimal, and all N drones are required to assemble within a given time T, thus resulting in the following distributed problem:

[0014]

[0015] Where, x i For the location of each drone, f i (x i ):R n →R is the local objective function of the UAV, f(x) is the global decision function, and x represents the position of the UAV under the condition of global optimum.

[0016] In step 1), the local objective function of each UAV must meet the following requirements:

[0017] Local objective function f i It is continuously differentiable and is m i - Strongly convex function, where m i >0 and satisfies the following formula: For gradient ▽f i It is M i -lipchitz, of which M i >0 and satisfy

[0018] In step 1), the communication relationship graph G between multiple unmanned aerial vehicles is strongly connected.

[0019] The goal of the distributed optimization control method in step 2) is to enable multiple UAVs to aggregate to the globally optimal position within a given time:

[0020] The following algorithm is applied to multiple drones:

[0021]

[0022] In the formula, t∈[0,T), the constant k∈(1,2),▽f i (x i (t) is f i (x i The gradient of (t) is a ij It is a component of the adjacency matrix A of the unbalanced directed graph G, x j (t) is a neighbor of drone i, y i (t) is x i The auxiliary variable z of (t) ii It is z i The i-th variable, α, β are positive parameters.

[0023] Transform a distributed optimization control method with a given time into a distributed optimization problem with an infinite time domain:

[0024] The upper bound of the assembly time of multiple UAVs is T. When t∈[0,T), the following time-domain coordinate mapping is used to transform the finite time domain t into the infinite time domain τ.

[0025]

[0026] The inverse transform is denoted as:

[0027]

[0028] Differentiating both sides of the above equation with respect to τ, we get...

[0029]

[0030] Therefore, based on the established relationship between the finite and infinite time domains, the following expression is derived within the infinite time domain τ:

[0031]

[0032] Let x = col(x1, x2, ..., x) n ), y = col(y1, y2…y n Z = diag(z) 11 ,…z NN ), z = col(z1, z2, ... z n ), ▽f=col(▽f1(x1),▽f2(x2)…▽f N (x N The above algorithm can be rewritten as follows:

[0033]

[0034] The method for proving in step 3) that the equilibrium point obtained by the distributed optimization control algorithm is the required optimal cluster point is as follows:

[0035] When y(0) = 0 Nn Then col(x) * ,y * ) is the equilibrium point of system (7) if and only if x * It is the optimal solution to problem (1), where

[0036] col(x * ,y * ) is the equilibrium point of system (7), therefore we have

[0037]

[0038] Because y(0) = 0 Nn , Therefore, there is Then it was introduced

[0039] We multiply the first expression of equation (8) on the left. Based on the results above, we can conclude that

[0040]

[0041] Therefore, the equilibrium point of the system is the optimal point.

[0042] The advantages and positive effects of this invention are:

[0043] 1) Unlike most existing multi-agent distributed optimization control algorithms, this invention considers the communication situation of UAV formation communication network as a weight-unbalanced directed graph, which is more in line with the communication environment of UAV formation when performing tasks.

[0044] 2) In algorithm design, instead of manually setting up cluster points, the algorithm intelligently obtains the optimal cluster points by calculating the objective function, making the algorithm more intelligent and more in line with the characteristics of UAV intelligence;

[0045] 3) Unlike current research on UAV assembly, this invention takes into account the key factor of assembly time, requiring that the assembly point not only be globally optimal, but also reach the optimal assembly point within a given time. Attached Figure Description

[0046] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. However, it should be understood that these drawings are designed for illustrative purposes only and are not intended to limit the scope of the present invention. Furthermore, unless specifically indicated, these drawings are intended only to conceptually illustrate the structural construction described herein and are not necessarily drawn to scale.

[0047] Figure 1 The initial position of the UAV provided in Embodiment 1 of the present invention;

[0048] Figure 2 The UAV flight trajectory x provided in Embodiment 1 of the present invention i1 ;

[0049] Figure 3 The UAV flight trajectory x provided in Embodiment 1 of the present invention i2 ;

[0050] Figure 4 The initial position of the UAV provided in Embodiment 2 of the present invention;

[0051] Figure 5 The UAV flight trajectory x provided in Embodiment 2 of the present invention i1 ;

[0052] Figure 6 The UAV flight trajectory x provided in Embodiment 2 of the present invention i2 . Detailed Implementation

[0053] First, it should be noted that the specific structure, features and advantages of the present invention will be described in detail below by way of examples. However, all descriptions are for illustrative purposes only and should not be construed as limiting the present invention in any way.

[0054] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other.

[0055] Example 1

[0056] The optimal aggregation algorithm for multiple UAVs in a weight-unbalanced directed communication network provided in this embodiment includes the following steps:

[0057] 1) When N drones gather in the same mission area, determine the local objective function of each drone and set the sum of all local objective functions as the total objective function of the entire drone swarm;

[0058] 2) To enable the UAV swarm to achieve optimal aggregation under the conditions of minimizing the overall objective function and the communication network graph being a weighted unbalanced directed graph, a corresponding distributed optimization control method is designed, and the equilibrium point of controlling the UAV swarm is obtained through this distributed optimization control method.

[0059] 3) Optimality analysis: Theoretically prove that the equilibrium point obtained by this distributed optimization control method is the required optimal cluster point;

[0060] 4) Convergence analysis: Prove that each drone can be controlled by the algorithm to gather at the optimal gathering point.

[0061] In step 1), when the drones assemble, the assembly point of the N drones is required to be globally optimal, and all N drones are required to assemble within a given time T, thus resulting in the following distributed problem:

[0062]

[0063] Where, x i For the location of each drone, f i (x i ):R n →R is the local objective function of the UAV, f(x) is the global decision function, and x represents the position of the UAV under the condition of global optimum.

[0064] In step 1), the local objective function of each UAV must meet the following requirements:

[0065] Local objective function f i It is continuously differentiable and is m i - Strongly convex function, where m i >0 and satisfies the following formula: For gradient ▽f i It is M i -lipchitz, of which M i >0 and satisfy

[0066] In step 1), the communication relationship graph G between multiple unmanned aerial vehicles is strongly connected.

[0067] The goal of the distributed optimization control method in step 2) is to enable multiple UAVs to aggregate to the globally optimal position within a given time:

[0068] The following algorithm is applied to multiple drones:

[0069]

[0070] In the formula, t∈[0,T), the constant k∈(1,2),▽f i (x i (t) is f i (x i The gradient of (t) is a ij It is a component of the adjacency matrix A of the unbalanced directed graph G, x j (t) is a neighbor of drone i, y i (t) is x i The auxiliary variable z of (t) ii It is z i The i-th variable, α, β are positive parameters.

[0071] Transform a distributed optimization control method with a given time into a distributed optimization problem with an infinite time domain:

[0072] The upper bound of the assembly time of multiple UAVs is T. When t∈[0,T), the following time-domain coordinate mapping is used to transform the finite time domain t into the infinite time domain τ.

[0073]

[0074] The inverse transform is denoted as:

[0075]

[0076] Differentiating both sides of the above equation with respect to τ, we get...

[0077]

[0078] Therefore, based on the established relationship between the finite and infinite time domains, the following expression is derived within the infinite time domain τ:

[0079]

[0080] Let x = col(x1, x2, ..., x) n ), y = col(y1, y2…y n Z = diag(z) 11 ,…zNN ), z = col(z1, z2, ... z n ), ▽f=col(▽f1(x1),▽f2(x2)…▽f N (x N The above algorithm can be rewritten as follows:

[0081]

[0082] It should be noted that z is introduced to eliminate the imbalance in the weighted directed graph. When i ≠ j, the initial value of z(0) satisfies z ij =0, z ii =1 (i = 1, 2, 3…N) This also ensures Z N -1 It is clearly defined.

[0083] The method for proving in step 3) that the equilibrium point obtained by the distributed optimization control algorithm is the required optimal cluster point is as follows:

[0084] When y(0) = 0 Nn Then col(x) * ,y * ) is the equilibrium point of system (7) if and only if x * It is the optimal solution to problem (1), where

[0085] col(x * ,y * ) is the equilibrium point of system (7), therefore we have

[0086]

[0087] Because y(0) = 0 Nn , Therefore, there is Then it was introduced

[0088] We multiply the first expression of equation (8) on the left. Based on the results above, we can conclude that

[0089]

[0090] Therefore, the equilibrium point of the system is the optimal point.

[0091] The convergence analysis method in step 4) is as follows:

[0092] When the local objective function satisfies the following inequality:

[0093]

[0094] in ξ min It is ξ=(ξ1,ξ2…ξ) N ) T The smallest eigenvector in the vector is M = max{M1, M2, ..., M}. N}, c = min{c1, c2, ..., c N},λ2(L N )yes middle The smallest eigenvalue when y(0)=0 Nn Then col(x,y) asymptotically converges to the system's equilibrium point col(x). * ,y * ),in x * This is the optimal solution to the problem.

[0095] According to the results of the literature: Therefore, as τ→∞, Z -1 →E -1 Therefore, for ease of analysis, a definition is established here. The system (7) is rewritten in the following form

[0096]

[0097] in

[0098]

[0099] The above system (12) can be regarded as a standard system. The perturbation system, where g(τ,ω)+u(τ) is the perturbation term of the system; first consider the stability of the standard system (31):

[0100]

[0101] set up Y = yy * , It can be rewritten as:

[0102]

[0103] Construct the following Lyapunov function

[0104]

[0105] The derivative of V1(X,Y) with respect to the above equation is:

[0106]

[0107] Define H = (X1, X2, ..., X...)N )∈R n×N D = (d1, d2, ... dn) N ), where d i =col(d i1 ,d i2 …d in )∈R n J = (1 N ,η2,η2…η N )∈R N×N Therefore, col i (H T )=d 1i 1 N +d 2i η2+…d Ni η N H T =JD T H = DJ T By Lemma 3, Define P = (Y1, Y2, ..., Y) N ), and combining with Lemma 4, we can obtain the following formula:

[0108]

[0109] review It can be obtained and define therefore

[0110]

[0111] For any According to YOUNG's inequality, we can obtain the following inequality:

[0112]

[0113] Therefore, there is

[0114]

[0115] in

[0116] According to the literature, it can be obtained that According to Lemma 2 and Lemma 3, the following inequality can be obtained:

[0117]

[0118] According to the formula: (xy) of a strongly convex function. T (▽f(x)-▽f(y))≥m i ||xy|| 2 X can be derived Tg(X)≥c||X|| 2 ,so

[0119] In summary

[0120]

[0121] definition

[0122]

[0123] The derivative of V2(X,Y) with respect to the above equation is:

[0124]

[0125] Because E = diag(ξ1, ξ2, ... ξ) N Therefore, we can obtain According to assumption 3, g(X) ≤ M||X||, and according to YOUNG's inequality...

[0126]

[0127] therefore

[0128]

[0129] definition

[0130]

[0131] The derivative of V3(X,Y) with respect to the above equation is:

[0132]

[0133] In summary

[0134]

[0135]

[0136] Therefore, it can be proved that inequality (10) is satisfied. According to the Lasalle invariance principle, when τ→∞, that is, when t→T, both X and Y tend to zero, that is, the solution (x,y) of system (13) tends to the equilibrium point of the system. y→y*, This is the optimal solution to problem (1). Next, consider the perturbation system g(τ,ω)+u(τ). According to the literature, there exist some positive constants δ,θ such that max|z ii -1 -ξ i -1 |≤δe -θτ , It can be concluded that g(τ,ω)+u(τ) is exponentially stable, and for bounded u(τ) such that The input state is stable. Therefore, after analyzing the stability of system (13), it can be seen from the literature that the trajectory of system (11) also converges to the equilibrium point. This proves that the distributed optimization algorithm designed in this paper can enable multiple UAVs to achieve optimal aggregation in a given time under a communication network of unbalanced directed graphs.

[0137] In this invention, it should be noted that R represents the set of real numbers, R n Let R represent n-dimensional Euclidean space. m×n Let I represent an m×n real matrix. n Represent an n-dimensional identity matrix, given n column vectors y1, y2, ..., y3. n ,col(y1,y2…y n ) represents the stacked vector y i (i = 1, 2, 3…n), diag() denotes a diagonal matrix, let matrix F ∈ R m×n F T Let F be the transpose of matrix F, ||F|| denotes the Euclidean norm of matrix F, and ||y|| denotes the Euclidean norm of vector y. The Kronecker product of matrices F and G

[12] .

[0138] When multiple drones assemble, a directed graph can be used to describe the communication between the drones during the assembly. The directed graph G = (V, E, A), where V represents the set of nodes in the directed graph G, and the values ​​in V represent the IDs of each drone. Let G be the edge set of a directed graph. When an edge (i,j)∈E, it means that when multiple drones are assembled, drone j can receive information sent by drone i, and drone j is called a neighbor of drone i. In the entire directed communication graph of drones, if all edges from point i to point j belong to E, then drone i can transmit information to drone j. If there is a path between any two drone nodes in the directed graph, then the directed graph of multi-drone communication is said to be strongly connected. A=[a ij ]∈R n×n Let G be the adjacency matrix, where a ii =0, when a ij When the value is greater than 0, it means that (i,j)∈E.

[0139] Indicates the inner neighbor of drone i Indicates the outer neighbor of drone i. and Represent the in-degree and out-degree of drone i, when d in (i)=d out(i) indicates that the directed graph is weight-balanced, and the Laplace matrix L of graph G is denoted as L. N =D in -A.

[0140] Lemma 1: If graph G is strongly connected, L N Let G be the Laplace matrix of graph G. Then the following holds:

[0141] (1) For L N Zero eigenvalue λ1(L N There exists a positive left eigenvector ξ = (ξ1, ξ2, ..., ξ). N ) T Furthermore, the left eigenvector ξ has the following properties: Therefore, ξ1, ξ2…ξ N A diagonal matrix of elements.

[0142] (2) It is positive semi-definite; let λ i for The eigenvalues ​​are sorted as follows:

[0143] 3. Convex Analysis

[0144] For a continuously differentiable function f: R n →R, ▽f denotes the gradient of f; f is a strongly convex function if and only if there exists m∈R>0, satisfying the following formula: The function f is C-lipchitz continuous if and only if Where C > 0.

[0145] Assume D∈R n×m vec(D) represents stacking the vectors of each column of D to obtain a column vector of nm. Based on this, two key lemmas hold:

[0146] Lemma 2: A∈R v×j , B∈R j×v Then tr(AB) = tr(BA) = (vec(A) T )) T vec(B)

[0147] Lemma 3: A∈R v×j , B∈R j×p , C∈R p×l Then there is

[0148] Specifically, in this embodiment, the optimal aggregation algorithm for multiple UAVs in a weight-imbalanced directed communication network is performed using the method of the present invention, as follows:

[0149] 1) Establish the initial position of each drone in the drone formation and convert the initial position into two-dimensional coordinates;

[0150] 2) Based on the status and position of each drone during drone assembly, establish a local objective function for each drone, and sum the local objective functions of each drone into a total objective function;

[0151] 3) Verify whether the objective function meets the algorithm requirements;

[0152] 4) Input the local objective function and initial position information into the algorithm, set the aggregation time, and solve the problem;

[0153] Write a program in Matlab 2016b to handle unmanned aerial vehicle (UAV) swarming. Assume the swarm consists of 10 UAVs. To make the simulation more intuitive, consider a distributed optimization problem of a multi-UAV system in a two-dimensional plane. The initial position of each UAV is randomly set. In this example, the x and y coordinates of each UAV are assumed to be arbitrarily chosen between 0 and 1. The initial coordinates of the 10 UAVs are as follows: Figure 1 As shown:

[0154] The communication network of 10 drones is a weighted unbalanced directed graph, and the adjacency matrix of its communication network is shown below:

[0155]

[0156] Next, we set the local objective functions for the 10 drones:

[0157] f1(x) = (x1 - 0.7) 2 +(x2+0.5) 2

[0158] f2(x)=(x1-0.7*2) 2 +(x² + 0.5*2) 2

[0159] f3(x) = (x1 - 0.7*3) 2 +(x2+0.5*3) 2

[0160] f4(x) = (x1 - 0.7*4) 2 +(x2+0.5*4) 2

[0161] f5(x) = (x1 - 0.7*5) 2 +(x2+0.5*5) 2

[0162] f6(x) = (x1 - 0.7*6)2 +(x2+0.5*6) 2

[0163] f7(x) = (x1 - 0.7 * 7) 2 +(x2+0.5*7) 2

[0164] f8(x) = (x1 - 0.7 * 8) 2 +(x2+0.5*8) 2

[0165] f9(x) = (x1 - 0.7*9) 2 +(x2+0.5*9) 2

[0166] f 10 (x) = (x1 - 0.7 * 10) 2 +(x2+0.5*10) 2

[0167] Clearly, f1(x) is quadratically differentiable, and f1(x) is also quadratically continuous. We substitute the initial coordinates of the UAV and the local objective function into the algorithm for simulation. This simulation sets the assembly time to 1 second. To make the results more intuitive, we represent the two-dimensional coordinates in one-dimensional form x. i1 and x i2 Show it, such as Figure 2 As shown:

[0168] Figure 2 and Figure 3 This demonstrates the trajectory and motion of the drone under algorithmic control, including... As the global optimum, it can be seen that the trajectories of the 10 drones converge to the globally optimal aggregation position within 1 second.

[0169] Example 2

[0170] In this embodiment, the optimal aggregation algorithm for multiple UAVs in a weighted unbalanced directed communication network is performed using the method of the present invention. In this example, the adjacency matrix of the communication network for 10 UAVs is the same as in Example 1, but the x and y coordinates of each UAV are changed to be arbitrarily selected between 0 and 5. The initial coordinates of the 10 UAVs are as follows: Figure 4 As shown:

[0171] The local objective functions of the 10 UAVs are referenced in the literature Chen Z, Ma J, Wang X, et al. Optimization of Continuous-time Multi-agent Systems over Directed Graph [C] / / IEEE 14th International Conference on Control and Automation. Anchorage, AK, USA: IEEE, 2018: 1-6, as follows:

[0172] f1(x) = 0.5*x1 2 +0.5*x2 2

[0173] f2(x) = 0.5*(x1+1) 2 +0.5*x2 2

[0174] f3(x)=0.5*x1 2 +0.5*(x2+1) 2

[0175] f4(x) = 0.5*(x1+1) 2 +0.5*(x2+1) 2

[0176] f5(x) = 0.25*x1 4 +0.25*x2 4

[0177] f6(x) = 0.25*(x1+1) 4 +0.25*x2 4

[0178] f7(x) = 0.25*x1 4 +0.25*(x2+1) 4

[0179] f8(x) = 0.25*(x1+1) 4 +0.25*(x2+1) 4

[0180] f9(x) = 0.5*x1 2 +0.5*x2 2

[0181] f 10 (x) = 0.5*(x1+1) 2 +0.5*x2 2

[0182] Clearly, in example 2, we obtain f. i (x) is twice continuously differentiable, f i (x) is also quadratic continuous. We input the pre-set initial coordinates of the UAV and the local objective function into the algorithm for simulation. The assembly time for this simulation is set to 15 seconds. To make the results more intuitive, we convert the two-dimensional coordinates into one-dimensional form x. i1 and x i2 Show it, such as Figure 5 , 6 As shown; Figure 5 and Figure 6 This demonstrates the trajectory and motion of the drone under algorithmic control, including... As the global optimum, it can be seen that the movement trajectories of the 10 drones converged to the globally optimal aggregation position within 15 seconds, thus verifying the effectiveness of the algorithm in this patent.

[0183] The above embodiments have provided a detailed description of the present invention, but the content described is only a preferred embodiment of the present invention and should not be considered as limiting the scope of the present invention. All equivalent variations and improvements made within the scope of the present invention should still fall within the patent coverage of the present invention.

Claims

1. An optimal aggregation method for multiple UAVs in a weight-imbalanced directed communication network, characterized in that, Includes the following steps: 1) When N drones gather in the same mission area, determine the local objective function of each drone and set the sum of all local objective functions as the overall objective function of the entire drone swarm; 2) To enable the UAV swarm to achieve optimal aggregation under the conditions of minimizing the overall objective function and the communication network graph being a weighted unbalanced directed graph, a corresponding distributed optimization control method is designed, and the equilibrium point of controlling the UAV swarm is obtained through this distributed optimization control method. 3) Optimality analysis: Theoretically, it is proven that the equilibrium point obtained by this distributed optimization control method is the optimal cluster point that satisfies the minimum of the overall objective function; 4) Convergence analysis: Prove that each drone can be controlled by the algorithm to gather at the optimal gathering point; The goal of the distributed optimization control method in step 2) is to enable multiple UAVs to aggregate to the globally optimal position within a given time: The following algorithm is applied to multiple drones: ； In the formula, ,constant , yes gradient, It is a weighted unbalanced directed graph adjacency matrix One component, It is a drone Neighbors yes Auxiliary variables, yes The One variable, It is a positive parameter; Transform a distributed optimization control method with a given time into a distributed optimization problem with an infinite time domain: The upper bound of the assembly time for multiple drones is ,when When the finite time domain t is transformed into the infinite time domain using the following time domain coordinate mapping, the following transformation is employed. : ； The inverse transform is denoted as: ； Align the left and right sides of the above equation Taking the derivative, we get: ； Therefore, based on the established relationship between the finite time domain and the infinite time domain, the following conclusions are drawn based on the infinite time domain. Expressions within: ； make , , , , The above algorithm is rewritten as follows: 。 2. The optimal aggregation method for multiple UAVs in a weight-unbalanced directed communication network as described in claim 1, characterized in that: In step 1), when the drones assemble, the assembly point of the N drones is required to be globally optimal, and all N drones are required to assemble within a given time T, thus resulting in the following distributed problem: ； in, For the location of each drone, : It is the local objective function of the UAV. As a global decision function This indicates the position of the drone under the condition of global optimum.

3. The optimal aggregation method for multiple UAVs in a weight-unbalanced directed communication network as described in claim 2, characterized in that: In step 1), the local objective function of each UAV must meet the following requirements: Local objective function It is continuously differentiable and is Strongly convex functions, where And satisfy the following formula: , ; For gradient yes of which, and satisfy .

4. The optimal aggregation method for multiple UAVs in a weight-unbalanced directed communication network as described in claim 3, characterized in that: In step 1), the communication relationship diagram between multiple unmanned aerial vehicles is shown. It is strongly connected.

5. The optimal aggregation algorithm for multiple UAVs in a weight-unbalanced directed communication network according to claim 1, characterized in that, The method for proving in step 3) that the equilibrium point obtained by the distributed optimization control algorithm is the required optimal cluster point is as follows: when ,but It is the equilibrium point of system (7) if and only if It is the optimal solution to problem (1), where ; It is the equilibrium point of system (7), therefore: ； because , Therefore, there is And then to introduce , We multiply the first expression of equation (8) on the left. Based on the results above, we can deduce that: ； Therefore, the equilibrium point of the system is the optimal point.

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