Unmanned aerial vehicle cluster distributed time-varying optimal formation tracking and verification method based on RISE-PI

Abstract

The invention discloses an unmanned aerial vehicle cluster distributed time-varying optimal formation tracking and verification method based on RISE-PI, and the method comprises the steps: obtaining the total number of unmanned aerial vehicle clusters, and defining the parameters of the unmanned aerial vehicle clusters; establishing a position loop dynamical model of the unmanned aerial vehicle and defining a time-varying formation pattern, a formation center and a cost function of the unmanned aerial vehicle; controller parameters are defined, a continuous distributed optimal formation tracking control protocol fusing RISE and PI algorithms is jointly constructed by combining parameters of the unmanned aerial vehicle cluster, neighbor states, a position loop dynamic model, a time-varying formation pattern and a cost function, and a time-varying optimal formation tracking task of the unmanned aerial vehicle cluster is achieved; the method can break through the technical bottleneck of an existing method in the aspect of fusing time-varying formation optimization and robust control, improves the flight stability, enables the flight distance of the unmanned aerial vehicle to be shortest, balances the task efficiency and the system performance, achieves the time-varying formation of the unmanned aerial vehicle cluster and the precise tracking task, and improves the reliability of the unmanned aerial vehicle cluster. And reliable theoretical basis and technical guarantee are provided for unmanned aerial vehicle cluster collaborative operation.

Description

Technical Field

[0001] The present invention relates to the field of unmanned aerial vehicle (UAV) cluster control technology, and in particular to a distributed time-varying optimal formation tracking method for UAV clusters based on RISE-PI. Background Art

[0002] With the increasing application of drone swarms in military reconnaissance, logistics distribution, and disaster relief, time-varying formation tracking technology, due to its excellent adaptability to dynamic environments, has become a core enabler for drone swarms to accomplish complex missions. Traditional formation control relies on fixed formations and centralized commands. However, centralized architectures are subject to single-point failure risks and communication bottlenecks, hindering the reliability and scalability of large-scale swarms. In this context, distributed optimization methods, which achieve global goals through local information exchange, can reduce communication overhead while ensuring optimal system performance, becoming a key technical direction for addressing swarm coordination. However, existing distributed algorithms for swarms employ discrete-time designs and rely on non-continuous control strategies such as symbolic functions. This can easily lead to control input jitter, resulting in reduced flight stability. Furthermore, traditional methods often separate formation tracking from performance optimization, making it difficult to achieve real-time decision-making and control coordination in dynamic environments. Therefore, a novel framework integrating continuous control and distributed optimization is urgently needed to achieve optimal performance for mission-oriented drone swarms while ensuring formation flexibility, thereby balancing stability and mission execution efficiency.

[0003] UAV swarm formation tracking control is a core technology for collaborative operations. With increasing mission complexity, time-varying formation tracking technology has become a research hotspot. Its core goal is to dynamically adjust the formation to adapt to environmental changes (e.g., target tracking, area coverage). Existing methods achieve formation reconfiguration by introducing time-varying reference trajectories or virtual leader models. However, most rely on global information exchange or preset trajectories, resulting in insufficient flexibility and difficulty responding to large-scale mission requirements in real time. Distributed optimization algorithms, which minimize the sum of local cost functions through local interactions, have become a key technical direction for achieving optimal performance in UAV swarms. Some studies have proposed distributed PI algorithms, which combine local gradients to control the error between neighboring states, achieving distributed performance optimization for UAV swarms. However, the cost function is a constant. For distributed optimization problems in multi-agent systems with time-varying cost functions, some studies have proposed edge-based adaptive protocols, using sign functions of neighboring errors to compensate for terms related to the time-varying cost function. Other studies have used state-dependent gains to address unbounded local cost functions, and have verified the theoretical results using quadrotor UAV experiments. However, these studies use sign functions, and discontinuous control methods can cause control input jitter. Furthermore, some studies have mentioned using gradient-based search methods to solve distributed time-varying quadratic optimization problems. Other studies have designed a time-varying formation control protocol based on a quadratic cost function for Euler-Lagrangian systems, but this is only applicable to solving quadratic optimization problems.

[0004] For stability analysis of distributed optimal formation tracking of UAV swarms, existing Lyapunov functionals designed based on tracking errors cannot directly analyze discontinuous sign functions. Furthermore, Lyapunov functions designed based on PI tracking errors are only applicable to time-invariant cost functions and cannot handle the analysis of time-varying optimal formations. Furthermore, the lack of analysis of the constant term in non-explicitly compensated time-varying cost functions makes the stability analysis less accurate when affected by the constant term. Therefore, a Lyapunov functional that eliminates sign function chattering and can analyze the constant term in time-varying cost functions is urgently needed to achieve stability analysis of distributed optimal formation tracking of UAV swarms. Summary of the Invention

[0005] The present invention provides a distributed time-varying optimal formation tracking and verification method for UAV clusters based on RISE-PI, so as to overcome the technical problem that the existing UAV cluster formation tracking process usually uses the UAV centralized formation control technology, which is prone to single point failure and waste of communication resources, can only solve quadratic optimization, and when using traditional control protocols, there will be control signal jitter caused by sign function.

[0006] In order to achieve the above object, the technical solution of the present invention is:

[0007] A distributed time-varying optimal formation tracking method for UAV swarms based on RISE-PI, including:

[0008] S1: Get the total number of drones in the swarm and define the parameters of the swarm based on the total number of drones;

[0009] S2: Establish the position loop dynamics model of UAVs and define the time-varying formation, formation center and cost function of UAVs;

[0010] S3: Define the controller parameters and the parameters and neighbor states of the UAV cluster, and jointly construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms with the position loop dynamics model, time-varying formation and cost function to achieve the time-varying optimal formation tracking task of the UAV cluster.

[0011] Furthermore, the total number of drones in the swarm is obtained, and the parameters of the drone swarm are defined based on the total number of drones, including:

[0012] S11. Get the total number M of drone clusters and define the vertex set of drone clusters based on the total number M. Edge set Ψ and neighbor set The specific steps are as follows:

[0013] S111: Define an undirected graph containing M drones in represents a set of vertices, is the edge set;

[0014] S112: Based on the edge set, the neighbor set of the i-th drone is defined as

[0015] S12. Define an undirected graph The adjacency matrix is If ψ ji ∈Ψ then a ij =1 otherwise a ij =0;

[0016] S13. Obtain the degree matrix of the drone cluster and define the Laplace communication matrix of the drone cluster based on the degree matrix and adjacency matrix of the drone cluster. The specific steps are as follows:

[0017] S131: Defining the degree matrix of a drone cluster in Represents node v i degree;

[0018] S132: Based on the degree matrix and adjacency matrix, the Laplace communication matrix of the drone cluster is defined as where l ij =-a ij ,i≠j, and

[0019] Define an undirected graph The correlation matrix is If the edge ψ ik Direction is from node v k To node v i , then w ik =-1, otherwise w ik =0, the new Laplace matrix expression is obtained based on the incidence matrix of the undirected graph:

[0020] Furthermore, a position loop dynamics model of the UAV is established and the time-varying formation, formation center and cost function of the UAV are defined, including:

[0021] S21. Define the position loop dynamics model of the i-th UAV, as shown in formula (1):

[0022]

[0023] in, is the state vector of the UAV on the lateral, longitudinal and altitude axes, is the control input vector of the UAV;

[0024] S22. Define the expected continuous and differentiable time-varying formation of the UAV swarm Define the local formation center r of the i-th UAV based on the UAV's state vector and time-varying formation formation i (t), as shown in formula (2),

[0025] r i (t) = p i (t)-h i (t) (2)

[0026] S23, based on the formation center, define the local cost function f of the i-th UAV obtained only based on its own state i (r i ,t), as shown in formula (3),

[0027] f i (r i ,t)=(r i +h i (t)-p i (0)) T (r i +h i (t)-pi (0)) (3)

[0028] S24. The sum of the cost functions of the drone cluster is defined based on the local cost function: and define It is a convex function and continuously differentiable. When all local formation centers approach the optimal formation center r * (t), the sum of the cost functions is minimized.

[0029] Furthermore, the controller parameters and the neighbor states of the UAV cluster are defined, and together with the position loop dynamics model, time-varying formation shape, and adjacency matrix, a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms is constructed, including:

[0030] S31, define the controller parameters β1, β2 and β3 and the neighbor state p of the UAV cluster j (t);

[0031] S32. For any bounded initial state p i (0), the definition formula for the time-varying formation tracking of the i-th UAV is constructed based on the position loop dynamics model of the i-th UAV, the time-varying formation formation and the optimal formation center, as shown in (4).

[0032]

[0033] S33, based on the definition of time-varying formation tracking of the i-th UAV, controller parameters β1, β2 and β3, and the neighbor state p of the UAV cluster j (t) and the adjacency matrix are used to construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms, as shown in formulas (5) and (6).

[0034]

[0035] Among them, u i represents the control protocol, i.e., the control input vector of the UAV, which is used to control the i-th UAV to form the desired formation. β1, β2, and β3 represent constant positive gains, and η i (t) represents the edge-based adjacent error sign integral, function ζ i (r i ,t) is based on the local cost function f i (r i ,t) defined by the cost function Hessian matrix and gradient combination function, as shown in formula (7),

[0036]

[0037] is the gradient, H i(r i ,t) is the Hessian matrix.

[0038] Based on the same inventive concept, a method for validating the effectiveness of the RISE-PI-based distributed time-varying optimal formation tracking method for UAV swarms is also proposed, including:

[0039] T1: Define the formation center tracking error, cost function constant error, and symbolic integral error based on the formation center, cost function, edge-based adjacent error symbolic integral, and communication matrix;

[0040] T2: Design a Lyapunov function. Based on the Lyapunov function and three errors, prove that the use of a continuous distributed optimal formation tracking control protocol can stabilize the UAV cluster system, track the formation center to the optimal formation center, minimize the sum of the cost functions of the UAV cluster, and enable the UAV cluster to form the designed formation.

[0041] Furthermore, the formation center tracking error, the cost function constant error, and the symbolic integral error are defined based on the formation center, the cost function, the edge-based adjacent error symbolic integral, and the communication matrix, including:

[0042] T11, based on the formation center, cost function, and edge-based adjacent error symbol integral, define the formation center tracking error, cost function constant error, and symbol integral error, as shown in formulas (8)-(10).

[0043]

[0044] in, represents the formation center tracking error, r k represents the center state of the k-th formation, χ i represents the cost function constant error, Denotes the symbolic integral error, Δ i and Δ k Expressed as a constant term in the i-th and k-th cost functions;

[0045] T12, based on the communication matrix, defines the compact form of the formation center tracking error and the symbolic integral error, as shown in formulas (11) and (12),

[0046]

[0047] in, Denote the compact form of the formation center tracking error and define represents the compact form of the symbolic integral error, χ represents the compact form of the cost function constant error, I3 represents the 3*3 identity matrix, sgn() is the symbolic function, α is a constant.

[0048] Furthermore, a Lyapunov function is designed. Based on the Lyapunov function and three errors, it is proved that the continuous distributed optimal formation tracking control protocol can stabilize the UAV swarm system, track the formation center to the optimal formation center, minimize the sum of the cost function of the UAV swarm, and enable the UAV swarm to form the designed formation, including:

[0049] T21, based on the formation center tracking error, defines the filtering error, as shown in formula (13),

[0050]

[0051] Substituting formula (11) into formula (13) redefines the filtering error as shown in formula (14),

[0052]

[0053] Calculate the derivative of the filtering error with respect to time, as shown in formula (15),

[0054]

[0055] Where, β2=αβ1; α is a constant;

[0056] T22, define the first Lyapunov function based on the formation center tracking error, filtering error and the derivative of filtering error with respect to time, as shown in formula (16),

[0057] V=V1+V2+V3 (16)

[0058] Among them, V represents the first Lyapunov function, which is used to verify that the UAV cluster can achieve the desired time-varying optimal formation tracking task based on the continuous distributed optimal formation tracking control protocol. V1 is the Lyapunov function that introduces the formation center tracking error, V2 is the Lyapunov function that introduces the filtering error, and V3 is the Lyapunov function that introduces the integral form of the formation center tracking error and the filtering error. V1, V2, and V3 are shown in formulas (17)-(19).

[0059]

[0060] Calculate the time derivative of V, as shown in formula (20),

[0061]

[0062] in, β2=αβ1、 0<θ<α, λ2 is The second smallest eigenvalue of θ, c1 and c2 are both constants;

[0063] Based on formula (20), we can get error e and Bounded, the UAV cluster system is stable; Based on the analysis of formula (20) based on Babalat's lemma, it can be obtained that the formation center tracking error tends to converge, that is, The symbolic integration error tends to converge, that is,

[0064] Combining formula (2), formula (8) and Available The error between the state vector of the UAV and the desired time-varying formation tends to converge, and the UAV cluster forms the desired time-varying formation. And can track the defined local formation center

[0065] T23. Define the second Lyapunov function based on the cost function, as shown in formula (21),

[0066]

[0067] Among them, the second Lyapunov function is used to prove that all formation centers can converge to the optimal formation center;

[0068] Derivative of the function Φ, as shown in formula (22),

[0069]

[0070] Based on Lyapunov's stability theory, it can be concluded that

[0071] As a convex function, it can be obtained that when t→∞, under the action of the continuous distributed optimal formation tracking control protocol Minimize, all formation centers r i Can track the optimal formation center r * ,Right now

[0072] Beneficial effects: The present invention provides a distributed time-varying optimal formation tracking method for UAV clusters based on RISE-PI, which has the following advantages:

[0073] 1. This paper addresses the formation control problem of UAV swarms by combining the Robust Integral of the Sign of the Error (RISE) strategy with the Proportional Integral (PI) algorithm to propose a continuous control protocol based on distributed optimization, namely the continuous distributed optimal formation tracking control protocol. This protocol significantly improves the control input jitter problem caused by traditional sign functions, overcomes the technical bottleneck of existing methods in integrating time-varying formation optimization and robust control, improves flight stability, and realizes time-varying formation and precise tracking tasks for UAV swarms.

[0074] 2. By designing the cost function of the distributed optimization algorithm and minimizing the cost function, a continuous control protocol is designed to minimize the UAV's flight distance, balancing mission effectiveness and system performance.

[0075] 3. By adopting a distributed control architecture, drones only need to obtain information about themselves and their neighbors, effectively avoiding the risk of single-point failure in the cluster and reducing communication consumption. It realizes the overall design of planning and control, simplifies the design process, and provides a reliable theoretical basis and technical guarantee for the collaborative operation of drone clusters, enabling drone clusters to achieve optimal performance with fewer communication resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0077] Figure 1 Flowchart of the distributed time-varying optimal formation tracking method for UAV clusters based on RISE-PI provided by the present invention;

[0078] Figure 2 Flowchart of the distributed time-varying optimal formation verification method for UAV clusters based on RISE-PI provided by the present invention;

[0079] Figure 3 The communication topology diagram of the drone cluster;

[0080] Figure 4 This is a schematic diagram of the three-dimensional optimal formation tracking results of the UAV cluster;

[0081] Figure 5 This is a schematic diagram of the two-dimensional optimal formation tracking results of the UAV cluster;

[0082] Figure 6 is a schematic diagram of the cost function curve;

[0083] Figure 7 It is a schematic diagram of the control law in the horizontal X-axis direction;

[0084] Figure 8 It is a schematic diagram of the control law in the longitudinal Y-axis direction;

[0085] Figure 9 Schematic diagram of the control law for the height Z-axis direction;

[0086] Figure 10 Schematic diagram of formation tracking error in the horizontal X-axis direction;

[0087] Figure 11 Schematic diagram of formation tracking error in the longitudinal Y-axis direction;

[0088] Figure 12 Schematic diagram of formation tracking error in the height Z-axis direction. DETAILED DESCRIPTION

[0089] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0090] This embodiment provides a distributed time-varying optimal formation tracking method for UAV clusters based on RISE-PI. Figure 1 Shown, including:

[0091] S1: Get the total number of drones in the swarm and define the parameters of the swarm based on the total number of drones;

[0092] S2: Establish the position loop dynamics model of UAVs and define the time-varying formation, formation center and cost function of UAVs;

[0093] S3: Define the controller parameters and the parameters and neighbor states of the UAV cluster, and jointly construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms with the position loop dynamics model, time-varying formation and cost function to achieve the time-varying optimal formation tracking task of the UAV cluster.

[0094] Specifically, the total number of drones in the swarm is obtained, and the parameters of the drone swarm are defined based on the total number of drones for control protocol design and stability analysis.

[0095] Secondly, a position loop dynamics model of the UAVs is established, and the time-varying formation, formation center, and cost function of the UAVs are defined. By designing the cost function of a distributed optimization algorithm and minimizing the cost function, a continuous control protocol is designed to minimize the UAV flight distance, balancing mission effectiveness and system performance.

[0096] Finally, the controller parameters and the parameters and neighbor states of the UAV cluster are defined, and a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms is constructed together with the position loop dynamics model, time-varying formation and cost function to achieve the time-varying optimal formation tracking task of the UAV cluster. By combining the Robust Integralof the Sign of the Error (RISE) strategy with the Proportional Integral (PI) algorithm, a continuous control protocol based on distributed optimization is proposed, which significantly improves the control input jitter problem caused by the traditional sign function, breaks through the technical bottleneck of the existing methods in integrating time-varying formation optimization and robust control, improves flight stability, and realizes the time-varying formation and precise tracking task of the UAV cluster.

[0097] In a specific embodiment, the total number of drones in a swarm is obtained, and the solution for defining the parameters of the drone swarm based on the total number of drones is:

[0098] S11. Get the total number M of drone clusters and define the vertex set of drone clusters based on the total number M. Edge set Ψ and neighbor set The specific steps are as follows:

[0099] S111: Define an undirected graph containing M drones in represents a set of vertices, is the edge set;

[0100] S112: Based on the edge set, the neighbor set of the i-th drone is defined as

[0101] S12. Define an undirected graph The adjacency matrix of If ψ ji ∈Ψ then a ij =1 otherwise a ij =0;

[0102] S13. Obtain the degree matrix of the drone cluster and define the Laplace communication matrix of the drone cluster based on the degree matrix and adjacency matrix of the drone cluster. The specific steps are as follows:

[0103] S131: Defining the degree matrix of a drone cluster in Represents node v i degree;

[0104] S132: Based on the degree matrix and adjacency matrix, the Laplace communication matrix of the drone cluster is defined as where l ij =-a ij ,i≠j, and

[0105] Define an undirected graph The correlation matrix is If the edge ψ ik Direction is from node v k To node v i , then w ik =-1, otherwise w ik =0, the new Laplace matrix expression is obtained based on the incidence matrix of the undirected graph:

[0106] In this scheme, graph theory is used to characterize the communication mode of multiple intelligent agents. It is necessary to first define the vertex set, edge set and communication matrix for control protocol design and stability analysis. At the same time, a distributed control architecture is adopted in the design process. The drone only needs to obtain information about itself and its neighbors, which effectively avoids the risk of single point failure of the cluster and reduces communication consumption. It realizes the overall design of planning and control, simplifies the design process, and provides a reliable theoretical basis and technical guarantee for the collaborative operation of drone clusters, enabling drone clusters to achieve optimal performance with fewer communication resources.

[0107] In a specific embodiment, the scheme for establishing the position loop dynamics model of the UAV and defining the time-varying formation formation, formation center and cost function of the UAV is:

[0108] S21. Define the position loop dynamics model of the i-th (i=1,…,M) UAV, as shown in formula (23):

[0109]

[0110] in, is the state vector of the UAV on the lateral, longitudinal and altitude axes, is the control input vector of the UAV;

[0111] S22. Define the expected continuous and differentiable time-varying formation of the UAV swarm Define the local formation center r of the i-th UAV based on the UAV's state vector and time-varying formation formation i (t), as shown in formula (24),

[0112] r i (t) = p i (t)-h i (t) (24)

[0113] S23, based on the formation center, define the local cost function f of the i-th UAV obtained only based on its own state i (r i ,t), as shown in formula (25),

[0114] f i (r i ,t)=(r i +h i (t)-p i (0)) T (r i +h i (t)-p i (0)) (25)

[0115] S24. The sum of the cost functions of the drone cluster is defined based on the local cost function: definition is convex and continuously differentiable. When achieving the optimal formation tracking task, all local formation centers r i (t) will approach a common optimal formation center r * (t), and the sum of cost functions is minimized if and only if hour, f i (r i ,t), the global minimum value of f i (r i ,t) is a convex and continuously differentiable function.

[0116] In this scheme, by designing the cost function of the distributed optimization algorithm and designing a continuous control protocol with the goal of minimizing the cost function, the UAV flight distance is minimized, balancing the mission effectiveness and system performance.

[0117] In a specific embodiment, the controller parameters and the parameters of the UAV cluster, the neighbor status are defined, and together with the position loop dynamics model, the time-varying formation formation and the cost function, a continuous distributed optimal formation tracking control protocol integrating the RISE and PI algorithms is constructed to achieve the time-varying optimal formation tracking task of the UAV cluster:

[0118] S31, define the controller parameters β1, β2 and β3 and the neighbor state p of the UAV cluster j (t);

[0119] S32. For any bounded initial state pi (0), the definition formula for the time-varying formation tracking of the i-th UAV is constructed based on the position loop dynamics model of the i-th UAV, the time-varying formation formation and the optimal formation center, as shown in formula (26).

[0120]

[0121] S33, based on the definition of time-varying formation tracking of the i-th UAV, controller parameters β1, β2 and β3, and the neighbor state p of the UAV cluster j (t) and the adjacency matrix are used to construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms, as shown in formulas (27) and (28).

[0122]

[0123]

[0124] Among them, u i represents the control protocol, i.e., the control input vector of the UAV, which is used to control the i-th UAV to form the desired formation. β1, β2, and β3 represent constant positive gains, and η i (t) represents the edge-based adjacent error sign integral, and its initial state satisfies Function ζ i (r i ,t) is based on the local cost function f i (r i ,t) defined by the cost function Hessian matrix and gradient combination function, as shown in formula (29),

[0125]

[0126] Among them, i (r i ,t) satisfies ζ i (r i ,t)=-αr i +Δ i (t) condition, where α>0, Δ i (t) is smooth with bounded first and second derivatives, is the gradient, H i (r i ,t) is a Hessian matrix, satisfying H i (r i ,t)=H j (r j ,t) conditions.

[0127] In this scheme, a continuous control protocol based on distributed optimization is proposed to solve the formation control problem of UAV swarms by combining the Robust Integral of the Sign of the Error (RISE) strategy with the Proportional Integral (PI) algorithm. This protocol significantly improves the control input jitter problem caused by traditional sign functions, breaks through the technical bottleneck of existing methods in integrating time-varying formation optimization and robust control, improves flight stability, and realizes time-varying formation and precise tracking tasks of UAV swarms.

[0128] In order to solve the problem that the traditional Lyapunov function cannot analyze the discontinuous sign function and the constant term of the time-varying cost function, resulting in low accuracy of stability analysis under the influence of the constant term, this embodiment also provides a method for verifying the effectiveness of the distributed time-varying optimal formation tracking method for UAV clusters based on RISE-PI, such as Figure 2 As shown, including:

[0129] T1: Define the formation center tracking error, cost function constant error, and symbolic integral error based on the formation center, cost function, edge-based adjacent error symbolic integral, and communication matrix;

[0130] T2: Design a Lyapunov function. Based on the Lyapunov function and three errors, prove that the use of a continuous distributed optimal formation tracking control protocol can stabilize the UAV cluster system, track the formation center to the optimal formation center, minimize the sum of the cost functions of the UAV cluster, and enable the UAV cluster to form the designed formation.

[0131] Specifically, the formation center tracking error, cost function constant error, and symbolic integral error are first defined based on the formation center, cost function, edge-based adjacent error symbolic integral, and communication matrix. Through the synergistic effect of the three errors, the stability, convergence, and robustness of the formation can be ensured.

[0132] Secondly, a Lyapunov function is designed. Based on the Lyapunov function and three errors, it is proved that the use of a continuous distributed optimal formation tracking control protocol can enable the formation center to track the optimal formation center, the sum of the cost functions of the UAV cluster is minimized, the UAV cluster can form the designed formation, and the UAV cluster system is stable; by designing a composite Lyapunov function, the robustness and continuity of formation tracking can be proved, and based on the convex optimization gradient Lyapunov function, it can be independently proved that the UAV cluster can converge to the optimal formation center, and the continuous distributed optimal formation tracking control protocol can achieve the tracking task.

[0133] In a specific embodiment, a scheme for defining the formation center tracking error, the cost function constant error, and the symbolic integral error based on the formation center, the cost function, the edge-based adjacent error symbolic integral, and the communication matrix is:

[0134] T11, based on the formation center, cost function, and edge-based adjacent error symbol integral, define the formation center tracking error, cost function constant error, and symbol integral error, as shown in formulas (30)-(32).

[0135]

[0136] in, represents the formation center tracking error, r k represents the center state of the k-th formation, χ i represents the cost function constant error, Denotes the symbolic integral error, Δ i and Δ k Expressed as a constant term in the i-th and k-th cost functions;

[0137] T12. Based on the communication matrix, the compact form of the formation center tracking error and the symbol integration error is defined as shown in formulas (33) and (34).

[0138]

[0139] in, Denote the compact form of the formation center tracking error and define represents the compact form of the symbolic integral error, χ represents the compact form of the cost function constant error, I3 represents the 3*3 unit matrix, sgn() is the symbolic function, α is a constant; because Δ i The first and second derivatives of (t) are bounded, so and Bounded, satisfied and

[0140] In this scheme, the stability, convergence and robustness of the formation can be ensured through the synergistic effect of the three errors.

[0141] In a specific embodiment, a Lyapunov function is designed. Based on the Lyapunov function and three errors, it is proved that the continuous distributed optimal formation tracking control protocol can stabilize the UAV cluster system, track the formation center to the optimal formation center, minimize the sum of the cost functions of the UAV cluster, and enable the UAV cluster to form the designed formation. The scheme is:

[0142] T21, based on the formation center tracking error, defines the filtering error, as shown in formula (35),

[0143]

[0144] because so From this we can deduce

[0145] Substituting formula (33) into formula (35) redefines the filtering error as shown in formula (36),

[0146]

[0147] Calculate the time derivative of the filtering error as shown in formula (37),

[0148]

[0149] Where, β2=αβ1; α is a constant;

[0150] T22, define the first Lyapunov function based on the formation center tracking error, filtering error and the derivative of filtering error with respect to time, as shown in formula (38),

[0151] V=V1+V2+V3 (38)

[0152] Among them, V represents the first Lyapunov function, which is used to verify that the UAV cluster can achieve the desired time-varying optimal formation tracking task based on the continuous distributed optimal formation tracking control protocol. V1 is the Lyapunov function that introduces the formation center tracking error, V2 is the Lyapunov function that introduces the filtering error, and V3 is the Lyapunov function that introduces the integral form of the formation center tracking error and the filtering error. V1, V2, and V3 are shown in formulas (39)-(41).

[0153]

[0154] Calculate the time derivative of V, as shown in formula (42),

[0155]

[0156]

[0157] in, β2=αβ1、 0<θ<α, λ2 is The second smallest eigenvalue of θ, c1 and c2 are both constants;

[0158] Based on formula (42), we can get error e and Bounded, the UAV cluster system is stable; Based on the analysis of formula (42) based on Babalat's lemma, it can be obtained that the formation center tracking error tends to converge, that is, The symbolic integration error tends to converge, that is,

[0159] Combining formula (24), formula (30) and Available The error between the state vector of the UAV and the desired time-varying formation tends to converge, and the UAV cluster forms the desired time-varying formation. And can track the defined local formation center

[0160] T23. Define the second Lyapunov function based on the cost function, as shown in formula (43),

[0161]

[0162] Among them, the second Lyapunov function is used to prove that all formation centers can converge to the optimal formation center;

[0163] Taking the derivative of the function Φ, as shown in formula (44),

[0164]

[0165] Based on Lyapunov's stability theory, it can be concluded that

[0166] As a convex function, it can be obtained that when t→∞, under the action of the continuous distributed optimal formation tracking control protocol Minimize, all formation centers r i Can track the optimal formation center r * ,Right now

[0167] Therefore, the effectiveness of the distributed control protocol of RISE and PI algorithms can be proved through the first Lyapunov function and the second Lyapunov function. The UAV cluster successfully realizes the time-varying optimal formation tracking task, and the formation tracking error tends to zero.

[0168] In this scheme, by designing a composite Lyapunov function, the robustness and continuity of formation tracking can be proved, and by using the convex optimization gradient Lyapunov function, it can be independently proved that the drone cluster can converge to the optimal formation center.

[0169] Example 1:

[0170] 1. Assume that 4 UAVs (M=4) jointly perform a time-varying formation tracking task, and their communication topology is as follows: Figure 3 As shown; undirected graph of 4 drones The adjacency matrix is

[0171] Define an undirected graph The correlation matrix is in Specifically expressed as:

[0172]

[0173] 2. Define the expected continuous and differentiable time-varying elliptical formation of the UAV cluster as

[0174] Design the initial state value of each drone to be p1(0)=[-1,5,3] T , p2(0)=[-3,-0.2,4] T , p3(0)=[2,3,7] T and p4(0)=[0,-1,2.5] T ;

[0175] According to formulas (25) and (26), the optimal formation center r is calculated * (t) = [-0.5, 1.7, 4.125] T;

[0176] 3. Define the parameters of the continuous distributed optimal formation tracking control protocol based on the RISE and PI algorithms, that is, β1 = 16, β2 = 16, β3 = 20 in formulas (28) and (29); η1(0) = [-0.1, 0.1, 0.1] T 、η2(0)=[0.1,-0.1,-0.1] T 、η3(0)=[-0.2,0.1,-0.1] T 、η4(0)=[0.2,-0.1,0.1] T ;

[0177] The three-dimensional and two-dimensional positions and trajectories of the drone cluster at t = 0s, t = 1s, t = 15s and t = 60s are as follows Figure 4-5 As shown, Figure 4 This is the three-dimensional optimal formation tracking result of the UAV cluster. Figure 5 is the two-dimensional optimal formation tracking result of the UAV cluster, Figure 4 (a) Figure 5 (a) is the tracking result of the drone cluster at 0s. Figure 4 (b) Figure 5 (b) is the tracking result of the drone cluster in 1s. Figure 4 (c) Figure 5 (c) is the tracking result of the drone cluster in 15 seconds. Figure 4 (d) Figure 5 (d) is the tracking result of the drone cluster in 60 seconds, in which the four drones and their trajectories are distinguished by four different colors, and the hexagonal star represents the calculated optimal formation center r * (t)=[-0.5,1.7,4.125] T ,from Figure 4 and Figure 5 It can be seen that within 60 seconds, the drone cluster can circle r * (t) rotation, achieving the elliptical formation tracking task;

[0178] The bounded total cost function curve is as follows Figure 6 As shown in the figure, the total cost function tends to be stable over time and can converge to a constant range. The change of the cost function over time within the constant range indicates that the UAV cluster can form a time-varying formation with the formation center as the center, and the UAV cluster can form a formation with the closest distance to the initial point;

[0179] The control law curves in three dimensions are as follows Figure 7-9 As shown, from Figure 7 、 Figure 8 and Figure 9 It can be seen that all control signals are bounded and no obvious jitter occurs;

[0180] Figure 10-12 is the error curve in the three directions of X, Y and Z axis. Figure 10 、 Figure 11 and Figure 12 It can be seen that the optimal formation tracking errors in all dimensions converge to zero within 60 seconds, which fully demonstrates that the drone cluster can accurately track the optimal formation center and stably maintain the preset elliptical formation.

[0181] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A distributed time-varying optimal formation tracking method for UAV swarms based on RISE-PI, characterized by: include: S1: Get the total number of drones in the swarm and define the parameters of the swarm based on the total number of drones; S2: Establish the position loop dynamics model of UAVs and define the time-varying formation, formation center and cost function of UAVs; S3: Define the controller parameters, combine the parameters of the UAV cluster, the neighbor status, the position loop dynamics model, the time-varying formation formation and the cost function to jointly construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms to achieve the time-varying optimal formation tracking task of the UAV cluster.

2. The distributed time-varying optimal formation tracking method for UAV swarm based on RISE-PI according to claim 1 is characterized in that: Get the total number of drones in the swarm and define the parameters of the swarm based on the total number of drones, including: S11. Obtain the total number M of the drone cluster, and define the vertex set v, edge set Ψ, and neighbor set of the drone cluster based on the total number M. The specific steps are as follows: S111: Define an undirected graph containing M drones in represents a set of vertices, is the edge set; S112: Based on the edge set, the neighbor set of the i-th drone is defined as S12. Define an undirected graph The adjacency matrix of If ψ ji ∈Ψ then a ij =1 otherwise a ij =0; S13. Obtain the degree matrix of the drone cluster and define the Laplace communication matrix of the drone cluster based on the degree matrix and adjacency matrix of the drone cluster. The specific steps are as follows: S131: Defining the degree matrix of a drone cluster in Represents node v i degree; S132: Based on the degree matrix and adjacency matrix, the Laplace communication matrix of the drone cluster is defined as where l ij =-a ij ,i≠j, and Define an undirected graph The correlation matrix is If the edge ψ ik Direction is from node v k To node v i , then w ik =-1, otherwise w ik =0, based on the incidence matrix of the undirected graph, the new Laplace matrix expression is obtained as 3. The distributed time-varying optimal formation tracking method for UAV swarm based on RISE-PI according to claim 2 is characterized in that: Establish a position loop dynamics model for UAVs and define their time-varying formation, formation center, and cost function, including: S21. Define the position loop dynamics model of the i-th UAV, as shown in formula (1): in, is the state vector of the UAV on the lateral, longitudinal and altitude axes, is the control input vector of the UAV; S22. Define the expected continuous and differentiable time-varying formation h of the UAV swarm i (t), Define the local formation center r of the i-th UAV based on the UAV's state vector and time-varying formation formation i (t), as shown in formula (2), r i (t)=p i (t)-h i (t) (2) S23, based on the formation center, define the local cost function f of the i-th UAV obtained only based on its own state i (r i ,t), as shown in formula (3), f i (r i ,t)=(r i +h i (t)-p i (0)) T (r i +h i (t)-p i (0)) (3) S24. The sum of the cost functions of the drone cluster is defined based on the local cost function: and define It is a convex function and continuously differentiable. When all local formation centers approach the optimal formation center r * (t), the sum of the cost functions is minimized.

4. The distributed time-varying optimal formation tracking method for UAV swarms based on RISE-PI according to claim 3 is characterized in that: Define the controller parameters and the parameters of the UAV cluster, the neighbor status, and jointly construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms with the position loop dynamics model, time-varying formation, and cost function, including: S31, define the controller parameters β1, β2 and β3 and the neighbor state p of the UAV cluster j (t); S32. For any bounded initial state p i (0), the definition formula for the time-varying formation tracking of the i-th UAV is constructed based on the position loop dynamics model of the i-th UAV, the time-varying formation formation and the optimal formation center, as shown in (4). S33, based on the definition of time-varying formation tracking of the i-th UAV, controller parameters β1, β2 and β3, and the neighbor state p of the UAV cluster j (t) and the adjacency matrix are used to construct a continuous distributed optimal formation tracking control protocol that integrates the RISE and PI algorithms, as shown in formulas (5) and (6). Among them, u i represents the control protocol, i.e., the control input vector of the UAV, which is used to control the i-th UAV to form the desired formation. β1, β2, and β3 represent constant positive gains, and η i (t) represents the edge-based adjacent error sign integral, function ζ i (r i ,t) is based on the local cost function f i (r i ,t) defined by the cost function Hessian matrix and gradient combination function, as shown in formula (7), is the gradient, H i (r i ,t) is the Hessian matrix.

5. A method for validating the effectiveness of the RISE-PI-based distributed time-varying optimal formation tracking method for UAV swarms according to claim 4, characterized in that: Includes: T1: Define the formation center tracking error, cost function constant error, and symbolic integral error based on the formation center, cost function, edge-based adjacent error symbolic integral, and communication matrix; T2: Design a Lyapunov function. Based on the Lyapunov function and three errors, prove that the use of a continuous distributed optimal formation tracking control protocol can stabilize the UAV cluster system, track the formation center to the optimal formation center, minimize the sum of the cost functions of the UAV cluster, and enable the UAV cluster to form the designed formation.

6. The method for validating the effectiveness of the RISE-PI-based distributed time-varying optimal formation tracking method for UAV swarms according to claim 5, characterized in that: The formation center tracking error, cost function constant error, and symbolic integral error are defined based on the formation center, cost function, edge-based neighbor error symbolic integral, and communication matrix, including: T11, based on the formation center, cost function, and edge-based adjacent error symbol integral, define the formation center tracking error, cost function constant error, and symbol integral error, as shown in formulas (8)-(10). in, represents the formation center tracking error, r k represents the center state of the k-th formation, χ i represents the cost function constant error, Denotes the symbolic integral error, Δ i and Δ k Expressed as a constant term in the i-th and k-th cost functions; T12, based on the communication matrix, defines the compact form of the formation center tracking error and the symbolic integral error, as shown in formulas (11) and (12), in, Denote the compact form of the formation center tracking error and define represents the compact form of the symbolic integral error, χ represents the compact form of the cost function constant error, I3 represents the 3*3 identity matrix, sgn() is the symbolic function, α is a constant.

7. The method for validating the effectiveness of the RISE-PI-based distributed time-varying optimal formation tracking method for UAV swarms according to claim 6, characterized in that: Design a Lyapunov function. Based on the Lyapunov function and three errors, prove that the continuous distributed optimal formation tracking control protocol can stabilize the UAV swarm system, track the formation center to the optimal formation center, minimize the sum of the cost function of the UAV swarm, and enable the UAV swarm to form the designed formation, including: T21, based on the formation center tracking error, defines the filtering error, as shown in formula (13), Substituting formula (11) into formula (13) redefines the filtering error as shown in formula (14), Calculate the derivative of the filtering error with respect to time, as shown in formula (15), Where, β2=αβ1; α is a constant; T22, define the first Lyapunov function based on the formation center tracking error, filtering error and the derivative of filtering error with respect to time, as shown in formula (16), V=V1+V2+V3 (16) Among them, V represents the first Lyapunov function, which is used to verify that the UAV cluster can achieve the desired time-varying optimal formation tracking task based on the continuous distributed optimal formation tracking control protocol. V1 is the Lyapunov function that introduces the formation center tracking error, V2 is the Lyapunov function that introduces the filtering error, and V3 is the Lyapunov function that introduces the integral form of the formation center tracking error and the filtering error. V1, V2, and V3 are shown in formulas (17)-(19). Calculate the time derivative of V, as shown in formula (20), in, β2=αβ1、 λ2 is The second smallest eigenvalue of c1 and c2 are both constants; Based on formula (20), we can get error e and Bounded, the UAV cluster system is stable; Based on the analysis of formula (20) based on Babalat's lemma, it can be obtained that the formation center tracking error tends to converge, that is, The symbolic integration error tends to converge, that is, Combining formula (2), formula (8) and Available The error between the state vector of the UAV and the desired time-varying formation tends to converge, and the UAV cluster forms the desired time-varying formation. And can track the defined local formation center T23. Define the second Lyapunov function based on the cost function, as shown in formula (21), Among them, the second Lyapunov function is used to prove that all formation centers can converge to the optimal formation center; Derivative of the function Φ, as shown in formula (22), Based on Lyapunov's stability theory, it can be concluded that As a convex function, it can be obtained that when t→∞, under the action of the continuous distributed optimal formation tracking control protocol Minimize, all formation centers r i Can track the optimal formation center r * ,Right now

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