Unmanned aerial vehicle cluster confrontation control method and system based on fractional order dynamics equation

Abstract

This invention provides a method and system for UAV swarm adversarial control based on fractional-order dynamic equations. The method includes: performing long-range correlation detection on historical trajectory data of the UAV swarm; reconstructing the state space of the UAV swarm based on long-range correlation characteristics and state data; constructing fractional-order game dynamic equations based on the state space; performing discrete dimensionality reduction on the equations to obtain a set of algebraic equations; solving the set of algebraic equations; generating flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium and issuing them to the UAV swarm to control the UAV swarm in game adversarial combat. This method overcomes the shortcomings of existing integer-order game models in describing memory effects, nonlocality, and anomalous diffusion, and achieves high-precision prediction and efficient solution of multidimensional nonlinear adversarial combat in complex dynamic environments. It significantly reduces computational complexity and fundamentally solves the problems of long-term prediction distortion and high-dimensional convergence difficulties in traditional models.

Description

Technical Field

[0001] This invention relates to the field of artificial intelligence technology, and in particular to a method and system for combating drone swarms based on fractional-order dynamic equations. Background Technology

[0002] In modern high-tech adversarial environments, such as dynamic adversarial scenarios involving large-scale drone swarms and coordinated encirclement, the behavioral patterns of the game participants are becoming increasingly complex. Traditional game-theoretic adversarial system modeling is primarily based on integer-order differential equations.

[0003] However, with the increasing complexity of adversarial environments, existing game models based on integer-order differential equations have gradually revealed many problems. For example, integer-order derivatives are local operators that rely solely on neighborhood information, leading to long-term time-domain prediction distortion in actual adversarial scenarios. Furthermore, in complex environments, there are long-range, leap-like interactions between individual UAVs, and the classical Laplace operator, based on the Gaussian diffusion assumption, struggles to fit anomalous diffusion under heavy-tailed distributions, resulting in significant deviations in the prediction of cluster aggregation and dispersion behaviors. Additionally, the increasing cluster size triggers the curse of state space dimensionality, causing traditional methods such as finite difference and finite element methods to have exponentially increasing computational costs, making it difficult to meet the demands of real-time adversarial scenarios. Summary of the Invention

[0004] This invention provides a method and system for UAV swarm adversarial control based on fractional-order dynamic equations, which solves the problems of long-term prediction distortion, large computational load, and inability to meet the millisecond-level real-time decision-making requirements in complex UAV swarm adversarial scenarios in existing technologies.

[0005] This invention provides a method for UAV swarm adversarial control based on fractional-order dynamic equations, applied to edge computing devices, comprising: Acquire status data and historical trajectory data of the drone swarm participating in the confrontation; Long-range correlation detection is performed on the historical trajectory data, and the state space of the UAV cluster is reconstructed based on the detected long-range correlation features and the state data. Based on the state space, a fractional-order game dynamic equation is constructed. The fractional-order game dynamic equation characterizes the memory effect through the time fractional derivative and the spatial nonlocal action through the spatial fractional operator. The fractional-order game dynamics equations are discretized and reduced in dimensionality to obtain a set of algebraic equations. The set of algebraic equations is solved, and flight control commands are generated based on the policy density evolution results corresponding to the solved Nash equilibrium. The flight control commands are then sent to the UAV swarm to control the UAV swarm to engage in game competition.

[0006] According to the present invention, a method for controlling unmanned aerial vehicle (UAV) swarms based on fractional-order dynamic equations is provided, wherein constructing fractional-order game dynamic equations based on the state space includes: Determine the probability density function corresponding to the drone cluster in the state space; Based on the probability density function, as well as the left and right fractional derivatives, the spatial fractional operator is constructed. Based on the memory effect parameters of the UAV swarm, the time fractional derivative is constructed; the memory effect parameters characterize the environmental disturbances and control delays experienced by the UAV swarm. Based on the time fractional derivative, the space fractional operator, and the interaction payoff term, the fractional game dynamics equation is determined; The interaction benefit term characterizes the adversarial impact of the drone swarm.

[0007] According to the present invention, a method for UAV swarm adversarial control based on fractional-order dynamic equations is provided, wherein the spatial fractional-order operator is constructed based on the probability density function, and the left and right fractional-order derivatives, comprising: Obtain the initial physical dimension characteristics and boundary condition characteristics of the drone swarm; When the initial physical dimension characteristics satisfy the preset initial value condition form, the left fractional derivative and the right fractional derivative are configured as Caputo fractional derivatives; When the boundary condition features exhibit singularity, the left fractional derivative and the right fractional derivative are configured as Riemann-Liouville fractional derivatives. Based on the configured left and right fractional derivatives, the probability density function is differentiated to obtain the spatial fractional operator.

[0008] According to the present invention, a method for UAV swarm adversarial control based on fractional-order dynamic equations is provided, wherein the fractional-order game dynamic equations are discretized and dimensionality reduced to obtain a system of algebraic equations, including: Select fractional Legendre polynomials or fractional Chebyshev polynomials as a family of fractional orthogonal basis functions, and determine the cutoff order of the family of fractional orthogonal basis functions. According to the truncation order, the fractional game dynamics equation is expanded into a series based on the fractional orthogonal basis function family; By using the Galerkin projection method, the expanded fractional game dynamics equations are projected onto a finite-dimensional subspace to obtain a system of algebraic equations.

[0009] According to the present invention, a method for UAV swarm adversarial control based on fractional-order dynamic equations is provided, wherein solving the system of algebraic equations includes: Initialize the strategy coefficient vector and damping factor to be determined in the system of algebraic equations; Determine the residual vector and Jacobian matrix of the algebraic equation system, and generate the Hessian matrix based on the Jacobian matrix; When the Hessian matrix is ​​not positive, the damping factor is increased to construct a correction matrix, and the update step size is determined based on the correction matrix and the residual vector. The strategy coefficient vector is then updated based on the update step size. When the residual norm corresponding to the updated policy coefficient vector is less than the convergence threshold, the policy density evolution result corresponding to the Nash equilibrium is determined based on the updated policy coefficient vector.

[0010] According to the present invention, a method for controlling a drone swarm based on fractional-order dynamic equations includes, in the step of issuing flight control commands to the drone swarm to control the drone swarm in a game-like confrontation, and further comprising: The actual flight trajectory of the drone swarm is obtained, and the expected flight trajectory of the drone swarm is determined based on the policy density evolution result corresponding to the Nash equilibrium. Determine the trajectory error between the expected flight trajectory and the actual flight trajectory; When the trajectory error is greater than the error threshold, the objective function is to minimize the trajectory error. The time fractional order of the time fractional derivative and the space fractional order of the space fractional operator are fine-tuned by using particle swarm optimization or genetic algorithm. The parameters in the system of algebraic equations are updated based on the finely adjusted time fractional order and space fractional order.

[0011] This invention also provides a drone swarm defense control system based on fractional-order dynamic equations, applied to edge computing devices, comprising: The data acquisition module acquires the status data and historical trajectory data of the drone swarm participating in the confrontation; The spatial reconstruction module is used to perform long-range correlation detection on the historical trajectory data and reconstruct the state space of the UAV cluster based on the detected long-range correlation features and the state data. The equation construction module is used to construct fractional-order game dynamic equations based on the state space. The fractional-order game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal actions through spatial fractional operators. The solution control module is used to perform discrete dimensionality reduction processing on the fractional-order game dynamics equations to obtain a set of algebraic equations, solve the set of algebraic equations, generate flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium, and send the flight control commands to the UAV cluster to control the UAV cluster to conduct game confrontation.

[0012] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and running on the processor, wherein the processor executes the computer program to implement the UAV swarm countermeasure control method based on fractional-order dynamic equations as described above.

[0013] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the UAV swarm countermeasure control method based on fractional-order dynamic equations as described above.

[0014] The present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the UAV swarm countermeasure control method based on fractional-order dynamic equations as described above.

[0015] The present invention provides a method and system for UAV swarm adversarial control based on fractional-order dynamic equations. By acquiring historical trajectory data of UAV swarms to extract long-range correlation features and reconstructing the state space, the method integrates time and space fractional-order calculus operators into the game dynamic equations for dimensionality reduction and solution. This overcomes the shortcomings of existing integer-order game models in describing memory effects, nonlocality, and anomalous diffusion, and achieves high-precision prediction and efficient solution of multidimensional nonlinear adversarial in complex dynamic environments. It significantly reduces computational complexity and fundamentally solves the problems of long-term time-domain prediction distortion and high-dimensional convergence difficulty of traditional models. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0017] Figure 1 This is a flowchart illustrating the UAV swarm countermeasure control method based on fractional-order dynamic equations provided by the present invention. Figure 2 This is a mathematical diagram illustrating the process of constructing the fractional-order game dynamics equations provided by this invention. Figure 3This is a schematic diagram of the structure of the UAV swarm countermeasure control system based on fractional-order dynamic equations provided by the present invention; Figure 4 This is a schematic diagram of the structure of the electronic device provided by the present invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0019] In modern high-tech adversarial environments, such as dynamic adversarial scenarios involving large-scale drone swarms and coordinated encirclement, the behavioral patterns of the game participants (such as the unmanned combat units of the red and blue sides) are becoming increasingly complex. Traditional game adversarial system modeling is mainly based on integer-order differential equations, such as classical replicator dynamics, Hamilton-Jacobi-Bellman equations, and parabolic partial differential equations, to describe the interaction and evolution of multiple parties.

[0020] However, with the increasing complexity of adversarial environments, existing game models based on integer-order differential equations have gradually revealed the following problems: On the one hand, integer derivatives, as local operators, rely only on neighborhood information at a given point. However, in actual drone warfare, the current decisions of the participants are often influenced by long-term historical states, such as tactical inertia and psychological lag. For example, the recovery process of drone swarms after being interfered with usually exhibits power-law decay characteristics rather than simple exponential decay. Integer-order models cannot accurately describe this long-range memory, leading to severe distortion in long-term predictions.

[0021] On the other hand, in complex terrain or topological environments, the interactions between individual UAVs are not limited to adjacent neighboring nodes, but exhibit long-range hopping phenomena, manifesting as anomalous diffusion. The traditional classical Laplace operator (second derivative) is based on the assumption of Gaussian diffusion, which cannot fit such heavy-tailed distributions, thus leading to significant deviations in the prediction of swarm aggregation and dispersion behavior.

[0022] On the other hand, as the scale of drone swarms and the number of participants increase, the curse of dimensionality in the state space becomes increasingly prominent. Traditional numerical solution methods, such as the finite difference method and the finite element method, suffer from dense mesh partitioning when dealing with high-dimensional game problems. This not only leads to an exponential increase in computational cost but also results in slow convergence speed, making it difficult to meet the needs of millisecond-level real-time game adversarial combat in drone swarms.

[0023] To address this, this invention provides a method for adversarial control of UAV swarms based on fractional-order dynamic equations. It aims to reconstruct a high-dimensional state space by detecting long-range correlations in historical trajectories, and to construct game dynamic equations using fractional-order calculus theory. The method utilizes time fractional derivatives to accurately characterize historical memory effects and spatial fractional operators to quantify spatial nonlocal interactions. Furthermore, discrete dimensionality reduction transforms the complex infinite-dimensional model into a system of easily tractable algebraic equations. This enables high-precision prediction of complex game adversarial processes and rapid, efficient solution of high-dimensional dynamic game Nash equilibria. It overcomes the shortcomings of existing integer-order models, such as long-term time-domain prediction distortion and the curse of dimensionality, significantly improving the adversarial control effectiveness of UAV swarms in complex environments.

[0024] Figure 1 This is a flowchart illustrating the UAV swarm adversarial control method based on fractional-order dynamic equations provided by this invention. This method is applied to edge computing devices. Figure 1 As shown, the method includes: Step 110: Obtain the status data and historical trajectory data of the drone swarm participating in the confrontation; Step 120: Perform long-range correlation detection on historical trajectory data, and reconstruct the state space of the UAV cluster based on the detected long-range correlation features and state data; Step 130: Construct fractional game dynamic equations based on the state space. The fractional game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal effects through spatial fractional operators. Step 140: Discretize and reduce the dimensionality of the fractional-order game dynamics equations to obtain a set of algebraic equations. Solve the set of algebraic equations and generate flight control commands based on the policy density evolution results corresponding to the Nash equilibrium obtained from the solution. Send the flight control commands to the UAV swarm to control the UAV swarm to engage in game confrontation.

[0025] Specifically, in real-world dynamic adversarial scenarios, edge computing devices need to acquire real-time status and historical trajectory data of the drone swarms participating in the adversarial process. These edge computing devices are computing platforms deployed at the edge of the adversarial environment network (such as those built into drone formation command nodes or high-performance edge computing gateways). Being closer to the data source ensures that the end-to-end latency from perceived data to policy issuance is extremely low, such as less than 10ms. The acquired data refers to the drone swarms participating in the adversarial process. For example, a red team drone swarm performing a capture mission and a blue team drone swarm performing an escape mission in a mountainous environment.

[0026] Edge computing devices use sensing modules such as airborne LiDAR, GPS (Global Positioning System) devices, and communication links to collect real-time status data of drone swarms. This data reflects information about physical and environmental variables at the current moment, such as the drone's three-dimensional spatial position, flight speed, and remaining battery power. It can also acquire historical trajectory data, which is a set of continuous position evolutions and flight routes over a period of time.

[0027] Next, considering that in actual combat, the current decisions of drone swarms are often influenced by long-term historical states (such as tactical inertia and psychological lag), this embodiment of the invention first performs long-range correlation detection on the historical trajectory data after acquisition. This long-range correlation detection analyzes whether there are statistical characteristics in the historical trajectory data where observations that are far apart still maintain a significant correlation. For example, the memory depth of historical trajectories can be assessed by calculating the statistical index of the time series using Hurst index analysis. If the detection index meets preset conditions, such as a Hurst index greater than 0.5, a significant memory effect is determined, and long-range correlation features characterizing the depth of memory / strength of historical influence are extracted.

[0028] After this, long-range correlation features can be fused with the current state data to reconstruct the state space of the drone swarm. Traditional state spaces typically only contain the current Markov state variables, while the reconstructed state space is a high-dimensional composite space with deep temporal memory. It not only reflects the current dynamic position of the drone swarm, such as where it is and what state it is in at this moment, but also contains the evolutionary trend of the drone swarm's historical trajectory, thus laying the foundation for establishing a dynamic game model that better conforms to physical laws.

[0029] Furthermore, fractional-order game dynamics equations can be constructed based on the state space. This process essentially involves establishing a mathematical model describing the evolution of the policy density of the participants. The model introduces the temporal fractional derivative, utilizing its property as a convolution integral operator that accumulates information from all past time points to accurately characterize the memory effect of UAV swarms caused by environmental disturbances and control delays. Simultaneously, spatial fractional-order operators are used to quantify spatial nonlocal actions, effectively overcoming the limitation of traditional Laplace operators that can only assume Gaussian diffusion. This allows for a good fit of the heavy-tailed distribution behaviors of UAVs in complex terrain, such as long-range hopping and secondary diffusion avoidance.

[0030] In a specific application scenario, such as the capture-escape game between red and blue teams, the fractional-order game dynamics equation can be expressed as a coupled fractional-order reaction-diffusion equation: For the Red Team (the pursuers), their position in space... probability density function at The evolution equation is: For the blue team (the escapee), their position in space... probability density function at The evolution equation is: in, The time fraction order is used to characterize the persistence (memory decay rate) of the memory effect (the memory effect caused by airflow disturbance and control delay in drone swarms), and can be 0.85. and The spatial fractional order is used to quantify the diffusion behavior characteristics of the red and blue sides under spatial nonlocal interactions, such as... The value of 1.6 represents the Red team's Levy flight search pattern (super-diffusion) in mountainous environments. The value of 1.2 represents the blue team's secondary diffusion avoidance behavior; and They are respectively about of Fractional derivatives of order space and order derivatives Fractional derivative of a space; and The red team is in The diffusion coefficient at the location and the blue square at The diffusion coefficient at that location; and Represents the interaction benefit coefficient at the time of encounter; and These parameters represent other relevant control or offset terms. Through these parameters, the equations can accurately recreate the true evolutionary dynamics of the UAV swarm.

[0031] Finally, since the constructed infinite-dimensional fractional dynamic equations are difficult to solve directly, they can be discretized and dimensionality reduced to obtain a system of algebraic equations. That is, numerical approximation methods, such as the Galerkin projection spectrum method based on a family of fractional orthogonal basis functions, are used to project the continuous, infinite-dimensional fractional dynamic equations into a finite-dimensional subspace. This transforms the equations, which originally faced the curse of dimensionality, into a system of easily computed finite-dimensional ordinary differential equations or algebraic equations. After the transformation, the transformed system of equations can be quickly optimized to find the Nash equilibrium point, i.e., a stable equilibrium solution where none of the participants can unilaterally change their strategies to obtain better returns. Based on the solved Nash equilibrium solution, the evolution results of the strategy density for each participant are analyzed, i.e., the optimal probability distribution evolution trajectory of each UAV in the spatiotemporal dimension.

[0032] Subsequently, the evolutionary result needs to be translated into specific, executable flight control commands. For example, it can guide the red team to generate a flight path with the maximum probability of encirclement, and guide the blue team to generate an escape path with the minimum expectation of being captured. Finally, the edge computing device sends these flight control commands to the control nodes of the drone swarm via wireless communication links, so as to direct the drone swarm to perform precise tactical maneuvers for evasion and cooperative combat in a real-world adversarial environment.

[0033] The UAV swarm adversarial control method based on fractional-order dynamic equations provided by this invention extracts long-range correlation features by acquiring historical trajectory data of UAV swarms and reconstructing the state space. It integrates time and space fractional-order calculus operators into the game dynamic equations for dimensionality reduction and solution, overcoming the shortcomings of existing integer-order game models in describing memory effects, nonlocality, and anomalous diffusion. It achieves high-precision prediction and efficient solution of multidimensional nonlinear adversarial in complex dynamic environments, significantly reduces computational complexity, and fundamentally solves the problems of long-term time-domain prediction distortion and high-dimensional convergence difficulty of traditional models.

[0034] Based on the above embodiments, step 130 includes: Determine the probability density function of the drone swarm in the state space; Based on the probability density function, as well as the left and right fractional derivatives, a spatial fractional operator is constructed. Based on the memory effect parameters of UAV swarms, a time fractional derivative is constructed; the memory effect parameters characterize the environmental disturbances and control delays experienced by UAV swarms. Based on the time fractional derivative, spatial fractional operator, and interactive payoff term, the fractional game dynamics equation is determined; the interactive payoff term characterizes the adversarial impact of the drone swarm.

[0035] Specifically, the process of constructing fractional-order game dynamics equations based on the state space can include: Figure 2 This is a mathematical principle diagram of the process of constructing the fractional-order game dynamics equations provided by the present invention, such as... Figure 2 As shown, after reconstructing the state space of the drone swarm, it is necessary to first determine the probability density function of the drone swarm within the state space. This function (denoted as ) ) used for dynamic characterization of the first Each participating party (such as the red team's encirclement cluster or the blue team's escape cluster) at time State space location The probability distribution or spatial density of the strategy under the given conditions.

[0036] Building upon this foundation, to overcome the limitation that traditional second-order derivatives can only describe local Gaussian diffusion, this embodiment of the invention constructs a spatial fractional operator by combining the integration intervals of the left and right fractional derivatives. Specifically, in complex terrain or adversarial networks, the interaction or search and detection of UAVs (such as the Levy Flight flight search mode) often exhibits bidirectional long-range hop phenomena. Therefore, this embodiment of the invention introduces a generalized fractional SL operator (denoted as SL) that includes bidirectional integration (left and right integrals). The operator can be represented as: In the formula, The left fractional derivative represents the nonlocal cumulative effect starting from the left side of the space; The right fractional derivative represents the nonlocal cumulative effect starting from the right side of the space; and The order of the spatial fraction (usually taken as 1000). (between), used to accurately characterize the nonlocal range and anomalous diffusion degree of spatial interactions; and These are the lower bounds (left boundary points) of the state space of the drone swarm. ) and upper bound in space (right boundary point) ); The diffusion coefficient function, Let be the potential well function. Through the combined effect of the fractional derivatives at both ends, this spatial fractional operator can fit well the heavy-tailed distribution and sub / super-diffusion avoidance behavior of UAV swarms in complex adversarial environments.

[0037] Meanwhile, in the time dimension, a fractional-time derivative can be constructed based on the memory effect parameters of the drone swarm. As mentioned earlier, the recovery process of a drone swarm after being disturbed is not a memoryless Markov process. The memory effect parameters here (such as the fractional-time derivative)... The range of values ​​is This is specifically designed to quantify the persistence of decision-making memory. Its physical meaning lies in characterizing the environmental disturbances and control delays experienced by drone swarms. For example, in actual mountain encirclement operations, the drone formation is affected by strong airflow disturbances and communication link control delays, causing the current movement strategy to heavily rely on past movement states. By setting reasonable memory effect parameters (such as...) The time fractional derivative can then be used. Accurately reproduce this long-range memory with power-law decay characteristics.

[0038] Finally, by integrating spatiotemporal characteristics with adversarial interaction logic, a fractional-order game dynamics equation can be constructed. The complete expression of this equation is: In the formula, This is an interactive benefit item, which includes the player's strategy. Strategies with other participants The nonlinear coupling relationship between them can characterize the adversarial impact of drone swarms, such as the zero-sum game and mutual constraints between the maximum encirclement probability of the red side and the minimum capture expectation of the blue side.

[0039] Thus, a generalized SL-type game dynamics equation that fully integrates the memory effect, spatial nonlocality, and nonlinear adversarial influence has been constructed.

[0040] In this embodiment of the invention, a refined time-space bi-fractional game dynamics equation is constructed by combining the probability density function, left and right fractional derivatives, and memory effect parameters. This gives the equation clear physical interpretability and overcomes the shortcomings of rigid parameters in traditional models and the inability of local operators to fit long-range interactions. It can adapt online and accurately reflect the anomalous diffusion characteristics of airflow disturbances, control delays, and complex terrain, providing a highly accurate underlying framework for predicting game strategies in high-dimensional complex dynamic environments.

[0041] Based on the above embodiments, a spatial fractional operator is constructed based on the probability density function, the left fractional derivative, and the right fractional derivative, including: Obtain the initial physical dimension characteristics and boundary condition characteristics of the drone swarm; When the initial physical dimensional characteristics satisfy the preset initial value condition form, the left fractional derivative and the right fractional derivative are configured as Caputo fractional derivatives; When the boundary condition features are singular, the left and right fractional derivatives are configured as Riemannian and Liouville fractional derivatives. Based on the configured left and right fractional derivatives, the probability density function is differentiated to obtain the spatial fractional operator.

[0042] Specifically, the process of constructing a spatial fractional operator based on the probability density function, the left fractional derivative, and the right fractional derivative can include: First, it is necessary to determine the initial physical dimension characteristics and boundary condition characteristics of the drone swarm. In a dynamic game-theoretic scenario, the initial physical dimension characteristics are the macroscopic variables with clear real-world physical dimensions that the drone swarm possesses in its initial state, such as the initial three-dimensional position and initial flight speed of the drone swarm as read in real time by sensors. The boundary condition characteristics reflect the edge attributes of the adversarial environment in which the drone swarm is located, such as the physical barriers encountered by drones in complex mountainous or canyon terrain, or the mathematical singularities that may exist under special constraints such as radar detection blind spots.

[0043] Next, the definition type of the fractional derivative will be determined and selected. Specifically, when the initial physical dimensions satisfy a preset initial value condition form—that is, when the initial state of the UAV swarm has definite physical dimensions, and the solution requires strictly maintaining the classical integer-order initial value condition form to facilitate direct substitution of real sensor measurement data—the edge computing device will configure both the left and right fractional derivatives as Caputo fractional derivatives. The advantage of using the Caputo definition is that the initial value form required for the Laplace transform is completely consistent with classical integer-order calculus, which perfectly matches the physical state quantities directly measured by the UAV sensors, thus avoiding the problem of the initial values ​​lacking practical physical meaning due to the introduction of a fractional-order model.

[0044] On the other hand, when the boundary condition features are singular, or when it is necessary to describe fractional eigenmodes at a purely mathematical level, the edge computing device will configure the left and right fractional derivatives as Riemann-Liouville fractional derivatives.

[0045] Because the Riemann-Liouville definition possesses inherent mathematical completeness in dealing with nonhomogeneous boundary conditions and spatial singularities, this configuration is further complemented by strict fractional integral boundary conditions, for example, using fractional integral operators. To constrain the spatial boundaries on both the left and right sides, its expression is: By using such fractional integral boundary conditions, the risk of computational divergence caused by complex terrain boundaries or network topological singularities can be eliminated.

[0046] Finally, based on the configured left and right fractional derivatives, the probability density function can be differentiated to obtain the final spatial fractional operator adapted to the current adversarial environment. This process essentially involves substituting the selected Caputo or Riemann-Liouville operator, chosen according to environmental characteristics, into the specific computational framework of the generalized SL operator, which includes both left and right bidirectional integrals, to perform nonlocal integration-differentiation operations on the probability density function, thereby completing the precise construction of the spatial fractional operator.

[0047] In this embodiment of the invention, by introducing an adaptive fractional derivative definition selection strategy for initial physical dimensions and boundary singularity characteristics, the physical interpretability advantage of the Caputo definition is combined with the mathematical completeness advantage of the Riemann-Liouville definition in handling singular boundaries. This endows the game dynamics equation with strong environmental adaptability and ensures the physical rigor and computational stability of UAV swarms when performing combat confrontation modeling under arbitrary complex terrain boundaries.

[0048] Based on the above embodiments, in step 140, the fractional-order game dynamics equations are discretized and dimensionality reduced to obtain a system of algebraic equations, including: Select fractional Legendre polynomials or fractional Chebyshev polynomials as fractional orthogonal basis function families, and determine the cutoff order of the fractional orthogonal basis function family. According to the truncation order, the fractional game dynamics equations are expanded into series based on the family of fractional orthogonal basis functions. By using the Galerkin projection method, the expanded fractional game dynamics equations are projected onto a finite-dimensional subspace to obtain a system of algebraic equations.

[0049] Specifically, since the constructed fractional-order game dynamics equations are typically infinite-dimensional partial differential equations, solving them using traditional finite difference methods or finite element methods often requires extremely dense meshing when dealing with fractional-order or high-dimensional problems. This leads to an exponential increase in computational cost, resulting in the curse of dimensionality in the state space. Therefore, to meet the requirements of high real-time adversarial capabilities for UAV swarms, this invention proposes a dimensionality reduction strategy based on fractional-order spectral methods for fractional-order game dynamics equations.

[0050] In detail, firstly, fractional Legendre polynomials or fractional Chebyshev polynomials can be selected as the family of fractional orthogonal basis functions, and the cutoff order of the family of fractional orthogonal basis functions is determined. Here, the family of fractional orthogonal basis functions, due to the orthogonality and completeness conferred by the self-adjointness of the generalized SL operator, can fit the singular properties of fractional calculus. Simultaneously, to achieve the optimal balance between computational accuracy and computational efficiency, the cutoff order is dynamically determined based on a preset accuracy threshold.

[0051] After determining the family of basis functions and its truncation order, the edge computing device performs a series expansion of the fractional-order game dynamics equations using the fractional-order orthogonal basis function family, according to the truncation order. This expands the originally continuous and infinite-dimensional probability density function. The approximate expansion is the sum of the products of a finite number of spatial basis functions and time coefficients. This expansion is: in, The selected fractional orthogonal basis functions; The time evolution coefficients to be solved; To truncate the order, The value of must ensure that the truncation error generated by approximation through a finite series is strictly limited to a controllable range, that is, it must satisfy the error norm in the weighted space. ; The preset accuracy threshold, For probability density function Expand and truncate to the front The finite-dimensional approximation function following the term; The weights in the weighted space are determined by orthogonal basis functions.

[0052] After this process, the problem of solving complex continuous functions can be transformed into the problem of solving a series of discrete time evolution coefficients.

[0053] After this, the expanded fractional-order game dynamics equations can be projected onto a finite-dimensional subspace using the Galerkin projection method, resulting in a system of algebraic equations. That is, by utilizing the orthogonality of orthogonal basis functions, the partial differential equations with approximate expansions are multiplied by each basis function. Due to the orthogonality, many cross terms are canceled out, thus efficiently projecting the infinite-dimensional partial differential game problem into a low-dimensional finite-dimensional subspace.

[0054] After projection discretization, the original equation is transformed into a problem concerning the expansion coefficients. , For nonlinear ordinary differential equation systems or low-dimensional algebraic equation systems, their typical matrix representation is as follows: in, This is the mass matrix generated by the inner product operation; This includes all the time evolution coefficients to be solved. The strategy coefficient vector; and These are the strategy coefficient vectors for the red team and the blue team (escapee), respectively. The fractional stiffness matrix generated by spatial differentiation, which can be used for acceleration. It is a nonlinear interaction term that includes multi-party strategy coupled game.

[0055] In this embodiment of the invention, the exponential convergence characteristic unique to the spectral method is fully utilized by using fractional orthogonal polynomials and Galerkin projection. Compared with the traditional finite difference method, the number of grid nodes required to achieve the same accuracy is reduced by an order of magnitude, which greatly reduces the computational complexity and resolves the dimensionality curse problem faced by high-dimensional dynamic games. The complex infinite-dimensional fractional partial differential equations are reduced to a system of low-dimensional algebraic equations that are easy to solve, thus clearing the computational obstacles for achieving millisecond-level real-time dynamic Nash equilibrium solutions for UAV swarms.

[0056] Based on the above embodiments, step 140 involves solving the system of algebraic equations, including: Initialize the strategy coefficient vector and damping factor to be determined in the system of algebraic equations; Determine the residual vector and Jacobian matrix of the system of algebraic equations, and generate the Hessian matrix based on the Jacobian matrix; When the Hessian matrix is ​​not positive, the damping factor is increased to construct a correction matrix, and the update step size is determined based on the correction matrix and the residual vector. The strategy coefficient vector is then updated based on the update step size. When the residual norm corresponding to the updated policy coefficient vector is less than the convergence threshold, the policy density evolution result corresponding to the Nash equilibrium is determined based on the updated policy coefficient vector.

[0057] Specifically, after successfully transforming an infinite-dimensional fractional partial differential equation into a finite-dimensional algebraic equation system using spectral methods, the key to achieving real-time control of UAV swarms lies in how to quickly and stably solve this system to obtain the optimal strategy. However, traditional heuristic algorithms are prone to getting trapped in local optima or facing divergence risks when solving high-dimensional dynamic games. Therefore, in this embodiment of the invention, an improved quasi-Newton iterative algorithm with an adaptive damping factor is designed for solving the algebraic equation system problem.

[0058] In detail, the first step is to initialize the strategy coefficient vector and damping factor to be determined in the system of algebraic equations. That is, for the The second iteration (initial) ), set the initial strategy coefficient vector of all participants as (This vector is the time evolution coefficient to be solved during the dimensionality reduction expansion process) (Composition), while setting the initial damping factor. This damping factor is used to adjust the search step size and direction in subsequent iterations to avoid out-of-bounds or divergence in the complex game payoff space.

[0059] After initialization, the residual vector and Jacobian matrix of the algebraic equation system need to be calculated. In each iteration step, the residual vector can be calculated based on the current strategy coefficient vector. This vector represents the degree of error between the current calculation result and the true Nash equilibrium point, and the Jacobian matrix is ​​calculated simultaneously. That is, the matrix of first-order partial derivatives of the error function with respect to each policy coefficient can reflect the gradient change trend. Subsequently, the Hessian matrix is ​​approximated using the Jacobian matrix. This matrix reflects the second-order curvature characteristics of the error surface.

[0060] Because drone games in complex adversarial environments often face non-convex payoff surfaces, the conventional Newton-Raphson method is prone to iterative divergence due to the loss of positive definiteness of the Hessian matrix. Therefore, this invention introduces a dynamic damping judgment mechanism, which continuously assesses the positive definiteness of the Hessian matrix. When the Hessian matrix is ​​not positive definite, the damping factor is actively increased, and the increased damping factor (denoted as ) can be used to resolve the issue. Construct the correction matrix. Specifically, this is done by introducing the identity matrix. The increased damping factor corrects the potentially singular or non-positive definite Hessian matrix into an absolutely positive definite correction matrix. .

[0061] Based on this, the update step size of the current iteration step can be obtained by solving the system of linear equations constructed from the correction matrix and the residual vector. And update the policy coefficient vector accordingly, i.e. This adaptive damping mechanism can accelerate convergence in regions with gentle gradients and ensure stable descent in regions with steep curvature or non-convexity.

[0062] Finally, a convergence test is required. Specifically, if the residual norm corresponding to the updated policy coefficient vector is less than the convergence threshold, the policy density evolution result corresponding to the Nash equilibrium is determined based on the updated policy coefficient vector. During this test, the policy change over two consecutive iterations can be monitored, and it can be verified whether the residual norm has decreased to the preset threshold. Below, that is If the condition is met, it means the iteration has precisely locked onto the target, and the current converged coefficient vector can be output. .

[0063] Subsequently, by substituting the optimal coefficient vector back into the aforementioned fractional order expansion, the optimal strategy distribution of each participant in the game confrontation can be reconstructed, that is, the strategy density evolution result corresponding to the Nash equilibrium, thereby guiding the generation of subsequent flight control commands.

[0064] In this embodiment of the invention, an improved quasi-Newton iterative algorithm with adaptive damping factor is used to integrate the Jacobian matrix with the damping correction mechanism. This effectively prevents the risk of divergence of the algorithm under high-dimensional non-convex payoff surfaces of multiple UAVs. It not only ensures the absolute stability and global convergence of the numerical solution, but also achieves rapid locking of the Nash equilibrium point. This avoids the defect of traditional heuristic algorithms being prone to getting trapped in local optima, and provides a powerful solution engine for real-time millisecond-level game decision-making of UAV swarms.

[0065] Based on the above embodiments, in step 140, flight control commands are sent to the drone swarm to control the drone swarm to engage in game-like combat, and the process further includes: The actual flight trajectory of the drone swarm is obtained, and the expected flight trajectory of the drone swarm is determined based on the policy density evolution results corresponding to the Nash equilibrium. Determine the trajectory error between the expected flight trajectory and the actual flight trajectory; When the trajectory error is greater than the error threshold, the objective function is to minimize the trajectory error. The time fractional order of the time fractional derivative and the space fractional order of the space fractional operator are fine-tuned by using particle swarm optimization or genetic algorithm. The parameters in the system of algebraic equations are updated based on the finely tuned time fractional order and space fractional order.

[0066] Specifically, in real-world dynamic adversarial environments, situations change rapidly. For example, a drone swarm might suddenly encounter unknown strong airflow disturbances, or the opponent might temporarily change their tactical strategies. Such environmental abrupt changes can alter dynamic characteristics, such as a sudden shift in spatial diffusion patterns from normal diffusion to hyper-diffusion. To enable the game theory model to possess online adaptive adjustment and environmental tracking capabilities, this embodiment of the invention further designs an online order optimization strategy based on closed-loop feedback.

[0067] In detail, after the flight control commands are issued and executed, the edge computing device continuously acquires the actual flight trajectory of the drone swarm and extracts the expected flight trajectory. Here, the actual flight trajectory refers to the real flight path and state changes in the physical space transmitted in real time by airborne sensors such as LiDAR and GPS after the drone swarm executes the flight control commands; the expected flight trajectory refers to the optimal spatial evolution path that the drone swarm should follow, predicted by the theoretical model based on the Nash equilibrium point (i.e., the optimal policy density evolution result) solved by the solution engine.

[0068] Once the expected flight trajectory and the actual flight trajectory are determined, the error can be assessed based on both, thus obtaining the trajectory error. For example, the trajectory error can be calculated by reassessing the trajectory at a set time period, such as every 5 seconds, and comparing the spatial distance deviation or state vector deviation between the theoretically predicted expected flight trajectory and the actual flight trajectory fed back by the sensor.

[0069] During the adversarial process, when the trajectory error exceeds a threshold, such as when the calculated trajectory error exceeds a preset tolerance threshold of 10%, it indicates that the parameters of the current game dynamics model can no longer accurately depict the adversarial environment after a sudden change. At this point, an order optimization mechanism will be triggered. Specifically, the edge computing device will use minimizing the trajectory error as the objective function and fine-tune the time fractional order of the time fractional derivative and the spatial fractional order of the spatial fractional operator using optimization algorithms with global optimization capabilities, such as particle swarm optimization and genetic algorithms.

[0070] In this process, fine-tuning the temporal fractional order aims to re-match the latest historical memory weights and information transmission speeds, such as to cope with sudden increases in communication latency; while fine-tuning the spatial fractional order is to requantify the nonlocal range and anomalous diffusion index of spatial interactions, such as to identify and match the long-range jump behavior suddenly adopted by the opponent to avoid encirclement. Particle swarm optimization algorithms and genetic algorithms continuously iterate and optimize within the allowed order range, ultimately finding the optimal combination of orders that makes the theoretical trajectory best fit the actual trajectory.

[0071] Finally, based on the fine-tuned time fractional order and space fractional order, the parameters in the system of algebraic equations are updated. Specifically, the optimal order output by the optimization algorithm is used. , , Substitute back into the system of algebraic equations to update the fractional stiffness matrix resulting from fractional derivatives. The updated system of algebraic equations is used to solve for the game equilibrium in the next time step, including underlying parameters.

[0072] In this embodiment of the invention, a closed-loop feedback and order optimization mechanism is introduced to achieve environmental adaptation. In practical applications, it can keenly and online identify sudden changes in the adversarial environment and the opponent's strategy shift, and automatically match the optimal fractional order model structure. This completely solves the problems of rigid parameters and poor adaptability of traditional models, and greatly enhances the robustness and long-term prediction accuracy of game strategies in uncertain and dynamic adversarial environments.

[0073] It should be noted that although the foregoing embodiments mainly elaborate on the control method provided by this invention in the context of a drone swarm encirclement-escape game scenario, the game adversarial system modeling and equilibrium solution architecture based on the fractional-order Sturm-Liouville (F-SL) equation constructed by this invention is essentially a general complex system control framework for handling high-dimensional nonlinearity, long-range correlation, and anomalous diffusion characteristics. Therefore, the core solution logic of this invention is not limited to drone flight control in physical space; it can also be applied to many other dynamic adversarial scenarios, such as active defense games in cyberspace and high-frequency trading adversarial scenarios in finance.

[0074] In proactive defense game scenarios in cyberspace (lateral movement defense within large data center intranets), the attackers and defenders become network attackers (such as advanced persistent threats) and automated network defense systems. To adapt to the specific characteristics of network topology, the variables within the state space are defined... The network topology is mapped to a set of network nodes, and the discrete network topology is embedded into a continuous mathematical interval for modeling using the eigenmap technique of the graph Laplacian matrix. Regarding operator selection, to adapt to the characteristics of network packet transmission, the spatial fractional-order operator is replaced with the Caputo-Fabrizio fractional-order derivative (which has a non-singular kernel), thus enabling a smoother and more accurate description of the bursty behavior prevalent in network traffic. In the design of interaction terms, the nonlinear interaction terms in the game dynamics equations... It includes key attack and defense parameters such as vulnerability exploitation success rate and defense patch deployment delay.

[0075] In the specific control process, edge computing devices first monitor the network traffic logs in real time, much like acquiring drone trajectory data, to keenly identify low-frequency abnormal scanning behaviors in the network. Then, based on the extracted spatiotemporal evolution characteristics, an attack propagation model based on the graph fractional-order SL equation is constructed, and the hybrid strategy Nash equilibrium of the attacker and defender is quickly solved. The Nash equilibrium result reveals the optimal infection path that the attacker might take, and the optimal node isolation and patch distribution strategy that the defender should adopt. Finally, this defense strategy is translated into specific execution instructions, automatically generating and distributing SDN (Software Defined Network) flow table rules to dynamically and accurately block high-risk links.

[0076] In simulated advanced persistent threat (APS) attacks, this fractional order theory-based method successfully identified low-frequency, slow attacks that traditional threshold-based IDS (intrusion detection systems) often miss. It not only strictly limited the attacker's lateral movement range to within 40%, but also greatly reduced the false positive rate and reduced unnecessary node isolation operations by 25%, thereby maximizing the continuity of network services while providing strong defense.

[0077] In high-frequency trading scenarios (order book games between stock market market makers and high-frequency traders), the game space transforms into the high-frequency volatile financial capital market, with various high-frequency trading algorithms as participants. To capture the unique long-range memory and extreme volatility patterns of the financial market, specific boundary and parameter configurations are required. Specifically, the time fractional order parameter setting... It is no longer a fixed value, but a dynamic clustering feature related to market volatility. This means that it will automatically decrease during periods of sharp market fluctuations and intense battles between bulls and bears. The value of is selected to enhance the model's memory weight of historical price trends; in the handling of boundary conditions, the conventional physical space absorption boundary is abandoned and a reflection boundary condition is adopted instead to simulate the price limit rules in the real stock market; at the same time, the objective function of solving the Nash equilibrium is reconstructed into a weighted sum of maximizing the Sharpe ratio (measuring risk-adjusted return) and minimizing inventory risk (measuring capital occupation risk).

[0078] In the actual trading process, Level-2 market data is input in real time at an extremely high frequency, and fractional-order operators are used to fit the irregular and anomalous diffusion characteristics exhibited in the order flow in real time. Then, using the constructed fractional-order SL model, the distribution of market buying and selling pressure within a very short future time window can be predicted in advance. Based on this high-precision pressure distribution prediction, the solution engine quickly solves for the current Nash equilibrium point, thus deriving the market maker's optimal pricing strategy (Bid-AskSpread) and the optimal order quantity. Finally, these strategies are directly connected to the exchange's API (Application Programming Interface) to execute high-frequency order placement and cancellation instructions with millisecond-level latency.

[0079] Extensive historical backtesting data shows that, under the rigorous testing of extreme market crashes or surges, the trading strategy generated by this invention significantly reduces the maximum drawdown by 15% compared to the classic option pricing and market-making strategy under the traditional Black-Scholes framework, while the annualized return increases by 8% against the trend. This effectively overcomes the fatal flaw of classic financial models that underestimate the fat-tailed distribution characteristics of the market due to the assumption of a Gaussian distribution.

[0080] To support the efficient solution and cross-scenario application of the aforementioned complex fractional-order game dynamics equations, and to ensure the real-time performance of the model in extremely dynamic environments, the method provided in this invention not only performs dimensionality reduction and adaptive optimization at the algorithm level, but also relies on a dedicated heterogeneous computing platform for its underlying physical implementation. Specifically, the method provided in this invention can be deployed in a high-performance edge computing gateway, and its general hardware architecture supporting the entire solution logic includes: (1) FPGA Acceleration Unit: After the game equations are discretized and reduced in dimension using the spectral method, a large number of matrix-vector multiplication operations are generated. The FPGA (Field Programmable Gate Array) acceleration unit is specifically used for parallel computation of the huge matrices generated after the discretization of fractional-order operators. At the hardware acceleration level, the Toeplitz structure property (i.e., a matrix structure with identical diagonal elements) of the fractional-order stiffness matrix is ​​utilized. Through hardware-level algorithms such as the Fast Fourier Transform, the computational complexity of matrix-vector multiplication is reduced from... Down to This breakthrough overcomes the computational bottleneck in solving high-dimensional dynamic games. (2) GPU cluster: Used to run intelligent optimization and verification tasks with huge concurrent computing requirements. When the adversarial environment changes suddenly and triggers the closed-loop feedback mechanism to perform online order optimization, the GPU (Graphics Processing Unit) cluster, with its concurrent floating-point operation capability of tens of thousands of stream processors, can run particle swarm optimization algorithm or genetic algorithm for global optimization at high speed, and at the same time undertake large-scale Monte Carlo simulation verification tasks, thereby ensuring the rapid output and absolute reliability of model fine-tuning parameters; (3) High-speed bus: As the neural hub for internal and external interactions within the edge computing gateway, the high-speed bus is responsible for connecting data from heterogeneous hardware (FPGA, GPU) at high speed. It eliminates data transfer congestion at the physical communication level, ensuring that the end-to-end latency of the entire closed loop, from the front-end sensor network collecting multi-source heterogeneous sensing data, to the calculation engine solving and generating Nash equilibrium strategies, and finally to the control commands being sent to the execution end (such as UAV flight control system, SDN controller, exchange API, etc.), is strictly less than 10ms, meeting the millisecond-level real-time game requirements in real adversarial environments.

[0081] This invention introduces fractional-order Sturm-Liouville theory, integrates time and space fractional-order calculus operators into the game dynamics framework, and supplements it with a hardware-software co-operated dimensionality reduction acceleration and adaptive optimization mechanism. This fundamentally improves the modeling dimensionality and solution efficiency of complex game adversarial systems, and solves the problem of dimensionality curse faced by traditional integer-order models in long-term time-domain prediction distortion and high-dimensional state space. It has extremely broad and far-reaching industrial application prospects in many fields involving high-dimensional nonlinearity and long-range memory characteristics, such as UAV swarm collaborative control, cyberspace security defense, and high-frequency quantitative trading in finance.

[0082] The UAV swarm countermeasure control system based on fractional-order dynamic equations provided by this invention is described below. The UAV swarm countermeasure control system based on fractional-order dynamic equations described below can be referred to and corresponded to the UAV swarm countermeasure control method based on fractional-order dynamic equations described above.

[0083] Figure 3 This is a schematic diagram of the structure of the UAV swarm defense control system based on fractional-order dynamic equations provided by the present invention, as shown below. Figure 3 As shown, this system can be applied to edge computing devices, and the system includes: Data acquisition module 310 acquires the status data and historical trajectory data of the drone swarm participating in the confrontation; The spatial reconstruction module 320 is used to perform long-range correlation detection on the historical trajectory data and reconstruct the state space of the UAV cluster based on the detected long-range correlation features and the state data. The equation construction module 330 is used to construct fractional-order game dynamic equations based on the state space. The fractional-order game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal actions through spatial fractional operators. The solution control module 340 is used to perform discrete dimensionality reduction processing on the fractional-order game dynamics equations to obtain a set of algebraic equations, solve the set of algebraic equations, generate flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium, and send the flight control commands to the UAV cluster to control the UAV cluster to conduct game confrontation.

[0084] The UAV swarm adversarial control system provided by this invention extracts long-range correlation features by acquiring historical trajectory data of UAV swarms and reconstructing the state space. It integrates time and space fractional calculus operators into the game dynamics equations for dimensionality reduction and solution, overcoming the shortcomings of existing integer-order game models in describing memory effects, nonlocality, and anomalous diffusion. It achieves high-precision prediction and efficient solution of multidimensional nonlinear adversarial in complex dynamic environments, significantly reducing computational complexity and fundamentally solving the problems of long-term time-domain prediction distortion and high-dimensional convergence difficulty of traditional models.

[0085] Based on the above embodiments, the equation construction module 330 is used for: Determine the probability density function corresponding to the drone cluster in the state space; Based on the probability density function, as well as the left and right fractional derivatives, the spatial fractional operator is constructed. Based on the memory effect parameters of the UAV swarm, the time fractional derivative is constructed; the memory effect parameters characterize the environmental disturbances and control delays experienced by the UAV swarm. Based on the time fractional derivative, the space fractional operator, and the interaction payoff term, the fractional game dynamics equation is determined; The interaction benefit term characterizes the adversarial impact of the drone swarm.

[0086] Based on the above embodiments, the equation construction module 330 is used for: Obtain the initial physical dimension characteristics and boundary condition characteristics of the drone swarm; When the initial physical dimension characteristics satisfy the preset initial value condition form, the left fractional derivative and the right fractional derivative are configured as Caputo fractional derivatives; When the boundary condition features exhibit singularity, the left fractional derivative and the right fractional derivative are configured as Riemann-Liouville fractional derivatives. Based on the configured left and right fractional derivatives, the probability density function is differentiated to obtain the spatial fractional operator.

[0087] Based on the above embodiments, the solution control module 340 is used for: Select fractional Legendre polynomials or fractional Chebyshev polynomials as a family of fractional orthogonal basis functions, and determine the cutoff order of the family of fractional orthogonal basis functions. According to the truncation order, the fractional game dynamics equation is expanded into a series based on the fractional orthogonal basis function family; By using the Galerkin projection method, the expanded fractional game dynamics equations are projected onto a finite-dimensional subspace to obtain a system of algebraic equations.

[0088] Based on the above embodiments, the solution control module 340 is used for: Initialize the strategy coefficient vector and damping factor to be determined in the system of algebraic equations; Determine the residual vector and Jacobian matrix of the algebraic equation system, and generate the Hessian matrix based on the Jacobian matrix; When the Hessian matrix is ​​not positive, the damping factor is increased to construct a correction matrix, and the update step size is determined based on the correction matrix and the residual vector. The strategy coefficient vector is then updated based on the update step size. When the residual norm corresponding to the updated policy coefficient vector is less than the convergence threshold, the policy density evolution result corresponding to the Nash equilibrium is determined based on the updated policy coefficient vector.

[0089] Based on the above embodiments, the system further includes a parameter update module, used for: The actual flight trajectory of the drone swarm is obtained, and the expected flight trajectory of the drone swarm is determined based on the policy density evolution result corresponding to the Nash equilibrium. Determine the trajectory error between the expected flight trajectory and the actual flight trajectory; When the trajectory error is greater than the error threshold, the objective function is to minimize the trajectory error. The time fractional order of the time fractional derivative and the space fractional order of the space fractional operator are fine-tuned by using particle swarm optimization or genetic algorithm. The parameters in the system of algebraic equations are updated based on the finely adjusted time fractional order and space fractional order.

[0090] Figure 4 An example is a schematic diagram of the physical structure of an electronic device, such as... Figure 4 As shown, the electronic device may include: a processor 410, a communications interface 420, a memory 430, and a communications bus 440, wherein the processor 410, the communications interface 420, and the memory 430 communicate with each other through the communications bus 440. The processor 410 can call logic instructions in the memory 430 to execute a drone swarm adversarial control method based on fractional-order dynamic equations. This method is applied to an edge computing device and includes: acquiring state data and historical trajectory data of the drone swarm participating in the adversarial process; performing long-range correlation detection on the historical trajectory data, and reconstructing the state space of the drone swarm based on the detected long-range correlation features and the state data; constructing fractional-order game dynamic equations based on the state space, wherein the fractional-order game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal actions through spatial fractional operators; performing discrete dimensionality reduction processing on the fractional-order game dynamic equations to obtain a system of algebraic equations, solving the system of algebraic equations, generating flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium, and issuing the flight control commands to the drone swarm to control the drone swarm to engage in game adversarial activities.

[0091] Furthermore, the logical instructions in the aforementioned memory 430 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0092] On the other hand, the present invention also provides a computer program product, which includes a computer program stored on a non-transitory computer-readable storage medium. The computer program includes program instructions, which, when executed by a computer, enable the computer to execute the UAV swarm adversarial control method based on fractional-order dynamic equations provided by the above methods. This method is applied to an edge computing device and includes: acquiring state data and historical trajectory data of a UAV swarm participating in the adversarial process; performing long-range correlation detection on the historical trajectory data, and reconstructing the state space of the UAV swarm based on the detected long-range correlation features and the state data; constructing fractional-order game dynamic equations based on the state space, wherein the fractional-order game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal actions through spatial fractional operators; performing discrete dimensionality reduction processing on the fractional-order game dynamic equations to obtain a system of algebraic equations; solving the system of algebraic equations; generating flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium; and issuing the flight control commands to the UAV swarm to control the UAV swarm to engage in game adversarial activities.

[0093] In another aspect, the present invention also provides a non-transitory computer-readable storage medium storing a computer program thereon. When executed by a processor, the computer program implements the UAV swarm adversarial control method based on fractional-order dynamic equations provided by the above methods. This method is applied to an edge computing device and includes: acquiring state data and historical trajectory data of the UAV swarm participating in the adversarial process; performing long-range correlation detection on the historical trajectory data, and reconstructing the state space of the UAV swarm based on the detected long-range correlation features and the state data; constructing fractional-order game dynamic equations based on the state space, wherein the fractional-order game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal actions through spatial fractional operators; performing discrete dimensionality reduction processing on the fractional-order game dynamic equations to obtain a system of algebraic equations; solving the system of algebraic equations; generating flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium; and issuing the flight control commands to the UAV swarm to control the UAV swarm to engage in game adversarial activities.

[0094] The system embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.

[0095] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0096] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for UAV swarm adversarial control based on fractional-order dynamic equations, characterized in that, Applications in edge computing devices include: Acquire status data and historical trajectory data of the drone swarm participating in the confrontation; Long-range correlation detection is performed on the historical trajectory data, and the state space of the UAV cluster is reconstructed based on the detected long-range correlation features and the state data. Based on the state space, a fractional-order game dynamic equation is constructed. The fractional-order game dynamic equation characterizes the memory effect through the time fractional derivative and the spatial nonlocal action through the spatial fractional operator. The fractional-order game dynamics equations are discretized and reduced in dimensionality to obtain a set of algebraic equations. The set of algebraic equations is solved, and flight control commands are generated based on the policy density evolution results corresponding to the solved Nash equilibrium. The flight control commands are then sent to the UAV swarm to control the UAV swarm to engage in game competition.

2. The UAV swarm countermeasure control method based on fractional-order dynamic equations according to claim 1, characterized in that, The construction of fractional-order game dynamics equations based on the state space includes: Determine the probability density function corresponding to the drone cluster in the state space; Based on the probability density function, as well as the left and right fractional derivatives, the spatial fractional operator is constructed. Based on the memory effect parameters of the UAV swarm, the time fractional derivative is constructed; the memory effect parameters characterize the environmental disturbances and control delays experienced by the UAV swarm. Based on the time fractional derivative, the space fractional operator, and the interaction payoff term, the fractional game dynamics equation is determined; The interaction benefit term characterizes the adversarial impact of the drone swarm.

3. The UAV swarm countermeasure control method based on fractional-order dynamic equations according to claim 2, characterized in that, The construction of the spatial fractional operator based on the probability density function, the left fractional derivative, and the right fractional derivative includes: Obtain the initial physical dimension characteristics and boundary condition characteristics of the drone swarm; When the initial physical dimension characteristics satisfy the preset initial value condition form, the left fractional derivative and the right fractional derivative are configured as Caputo fractional derivatives; When the boundary condition features exhibit singularity, the left fractional derivative and the right fractional derivative are configured as Riemann-Liouville fractional derivatives. Based on the configured left and right fractional derivatives, the probability density function is differentiated to obtain the spatial fractional operator.

4. The UAV swarm countermeasure control method based on fractional-order dynamic equations according to any one of claims 1 to 3, characterized in that, The discrete dimensionality reduction of the fractional-order game dynamics equations yields a system of algebraic equations, including: Select fractional Legendre polynomials or fractional Chebyshev polynomials as a family of fractional orthogonal basis functions, and determine the cutoff order of the family of fractional orthogonal basis functions. According to the truncation order, the fractional game dynamics equation is expanded into a series based on the fractional orthogonal basis function family; By using the Galerkin projection method, the expanded fractional game dynamics equations are projected onto a finite-dimensional subspace to obtain a system of algebraic equations.

5. The UAV swarm countermeasure control method based on fractional-order dynamic equations according to any one of claims 1 to 3, characterized in that, Solving the system of algebraic equations includes: Initialize the strategy coefficient vector and damping factor to be determined in the system of algebraic equations; Determine the residual vector and Jacobian matrix of the algebraic equation system, and generate the Hessian matrix based on the Jacobian matrix; When the Hessian matrix is ​​not positive, the damping factor is increased to construct a correction matrix, and the update step size is determined based on the correction matrix and the residual vector. The strategy coefficient vector is then updated based on the update step size. When the residual norm corresponding to the updated policy coefficient vector is less than the convergence threshold, the policy density evolution result corresponding to the Nash equilibrium is determined based on the updated policy coefficient vector.

6. The UAV swarm countermeasure control method based on fractional-order dynamic equations according to any one of claims 1 to 3, characterized in that, The step of issuing flight control commands to the drone swarm to control the drone swarm in a game-like confrontation further includes: The actual flight trajectory of the drone swarm is obtained, and the expected flight trajectory of the drone swarm is determined based on the policy density evolution result corresponding to the Nash equilibrium. Determine the trajectory error between the expected flight trajectory and the actual flight trajectory; When the trajectory error is greater than the error threshold, the objective function is to minimize the trajectory error. The time fractional order of the time fractional derivative and the space fractional order of the space fractional operator are fine-tuned by using particle swarm optimization or genetic algorithm. The parameters in the system of algebraic equations are updated based on the finely adjusted time fractional order and space fractional order.

7. A UAV swarm countermeasure control system based on fractional-order dynamic equations, characterized in that, Applications in edge computing devices include: The data acquisition module acquires the status data and historical trajectory data of the drone swarm participating in the confrontation; The spatial reconstruction module is used to perform long-range correlation detection on the historical trajectory data and reconstruct the state space of the UAV cluster based on the detected long-range correlation features and the state data. The equation construction module is used to construct fractional-order game dynamic equations based on the state space. The fractional-order game dynamic equations characterize the memory effect through time fractional derivatives and spatial nonlocal actions through spatial fractional operators. The solution control module is used to perform discrete dimensionality reduction processing on the fractional-order game dynamics equations to obtain a set of algebraic equations, solve the set of algebraic equations, generate flight control commands based on the policy density evolution results corresponding to the solved Nash equilibrium, and send the flight control commands to the UAV cluster to control the UAV cluster to conduct game confrontation.

8. An electronic device comprising a memory, a processor, and a computer program stored in the memory and running on the processor, characterized in that, When the processor executes the computer program, it implements the UAV swarm countermeasure control method based on fractional-order dynamic equations as described in any one of claims 1 to 6.

9. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the UAV swarm countermeasure control method based on fractional-order dynamic equations as described in any one of claims 1 to 6.

10. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the UAV swarm countermeasure control method based on fractional-order dynamic equations as described in any one of claims 1 to 6.

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