A method for aircraft trajectory prediction based on deep Koopman operator

Abstract

This invention discloses an aircraft trajectory prediction method based on deep Koopman operators. By combining deep learning with Koopman operator theory, a three-layer network is constructed to automatically extract Koopman features. Four physical constraints are introduced to improve the model's generalization ability, realizing a three-stage prediction strategy and dual-channel online learning. This solves the problems of low prediction accuracy, poor physical consistency, and weak adaptability of traditional methods in high-maneuverability flight scenarios, achieving high-precision aircraft trajectory prediction, providing reliable target information for the seeker system, and significantly improving the interception success rate.

Description

Technical Field

[0001] This invention belongs to the field of aircraft technology, specifically relating to an aircraft trajectory prediction method based on the depth Koupman operator. Background Technology

[0002] In modern intelligent air traffic management systems, aircraft trajectory prediction is a crucial step in route planning and optimization, directly impacting airspace utilization efficiency and flight safety. Traditional aircraft trajectory prediction methods mainly fall into the following categories:

[0003] 1. Physical model-based methods: These methods rely on accurate physical dynamic models, establishing a set of equations of motion for the aircraft using Newton's equations of motion and aerodynamic principles to predict its future trajectory. However, these methods require a large amount of prior knowledge and accurate flight parameters, and the model error is relatively large under high-maneuverability flight conditions, making trajectory prediction for unknown aircraft particularly difficult.

[0004] 2. Traditional data-driven methods: Statistical prediction methods, represented by Kalman filtering, perform well for linear or weakly nonlinear systems, but their prediction accuracy drops significantly when faced with highly nonlinear and highly maneuverable flight scenarios.

[0005] 3. Conventional machine learning methods: Although algorithms such as support vector machines can learn patterns from historical data, they are difficult to effectively capture the temporal dynamics and nonlinear characteristics of aircraft trajectories, and the error accumulation is large in long-term prediction.

[0006] 4. Traditional deep learning methods: Although deep learning methods, represented by Long Short-Term Memory (LSTM) networks, can process time-series data, they have limited understanding of physical constraints and are prone to producing prediction results that do not conform to the laws of flight dynamics.

[0007] The main limitations of the above methods are: first, it is difficult to simultaneously take into account the interpretability of the physical model and the adaptability of data-driven approaches; second, the computational complexity is high when dealing with high-dimensional nonlinear dynamic systems; and third, the consistency between the prediction results and the actual flight physics is poor. These problems are particularly prominent in high-speed and high-maneuverability flight scenarios.

[0008] Koopman operator theory provides a linear representation method for complex nonlinear dynamical systems, allowing them to be analyzed by mapping nonlinear systems to a high-dimensional linear space. However, traditional Koopman methods face challenges in practical applications, such as difficulty in choosing observation functions and high computational complexity. Summary of the Invention

[0009] To overcome the shortcomings of existing technologies, this invention provides an aircraft trajectory prediction method based on deep Koopman operators. By combining deep learning with Koopman operator theory, a three-layer network is constructed to automatically extract Koopman features. Four physical constraints are introduced to improve the model's generalization ability, realizing a three-stage prediction strategy and dual-channel online learning. This solves the problems of low prediction accuracy, poor physical consistency, and weak adaptability of traditional methods in high-maneuverability flight scenarios, achieving high-precision aircraft trajectory prediction, providing reliable target information for the seeker system, and significantly improving the interception success rate.

[0010] The technical solution adopted by this invention to solve its technical problem is as follows:

[0011] Step 1: Trajectory data acquisition and preprocessing;

[0012] Step 2: Deep Koopman model construction;

[0013] Step 3: Physically constrained deep Koopman model;

[0014] Step 4: Adaptive multi-step prediction strategy;

[0015] Step 5: Online learning and model updates.

[0016] Preferably, step 1 specifically comprises:

[0017] Step 1-1: Data Acquisition: Obtain the position, velocity, and acceleration status information of the target aircraft through multi-sensor fusion technology to form the original trajectory sequence X. raw ={x1,x2,...,x T},in Let d represent the d-dimensional state vector at time t, where t = 1…T;

[0018] Step 1-2: Noise Filtering: The Unscented Kalman Filter (UKF) method is used to filter noise from the original data. During the filtering process, the state covariance matrix Q is set as a diagonal matrix with diagonal elements [0.01, 0.01, 0.01, 0.05, 0.05, 0.05, 0.1, 0.1, 0.1]. The observation noise covariance matrix R is also set as a diagonal matrix with diagonal elements [0.1, 0.1, 0.1]. The filtered dataset X is obtained. filtered ;

[0019] Steps 1-3: Data Completion: To address data loss caused by sensor occlusion or malfunction, a cubic spline interpolation algorithm is used for data completion. The spline smoothing parameter is set to 0.8 to generate a complete trajectory sequence S. complete ;

[0020] Steps 1-4: Feature Extraction: Extracting position coordinates (po) from the complete trajectory x ,poy ,po z ), velocity (v) x ,v y ,v z ) and acceleration (a x ,a y ,a z Construct a feature vector set S feature ;

[0021] Steps 1-5: Data Standardization: Standardize the feature data to eliminate the influence of differences in the units of different features, resulting in a standardized dataset. where X i ={x i ,x i+1 ,...,x i+p-1 Let} be an input sequence of length p, and Y i ={x i+p ,x i+p+1 ,...,x i+p+q-1} represents the target sequence of length q; N represents the total number of samples in the dataset.

[0022] Preferably, step 2 specifically comprises:

[0023] Step 2-1: Encoder Network Design: Constructing the encoder network Φ θ : Where θ represents the network parameters, d is the original state space dimension, and k is the Koopman feature space dimension; the encoder network adopts a multilayer perceptron (MLP) structure.

[0024] Step 2-2: Decoder Network Design: Constructing the Decoder Network Ψ φ : Where φ represents the network parameters used to map the state of the Koopman feature space back to the original state space;

[0025] Steps 2-3: Koopman operator approximation: Constructing the Koopman matrix Where ω is a learnable parameter used to linearly advance the system state in the Koopman feature space;

[0026] Steps 2-4: Network Training: Define the total loss function L total It includes the following parts:

[0027] Reconstruction loss:

[0028] Predicted loss:

[0029] Linear consistency loss:

[0030] Regularization loss: L reg =0.001‖θ‖ 2 +0.001‖φ‖ 2 +0.0005‖ω‖ 2 ;

[0031] Total loss function: L total =0.3L recon +0.5L pred +0.2L lin +L reg ;

[0032] Minimize L using the Adam optimizer total The trained encoder network Φ is obtained by updating the parameters θ, φ, and ω through backpropagation. θ Decoder Network Ψ φ and the Koopman matrix K ω .

[0033] Preferably, step 3 specifically comprises:

[0034] Step 3-1: Definition of Flight Dynamics Constraints: Based on the principles of aircraft flight dynamics, define a set of physical constraints C. phys ={C1,C2,C3,C4}, including:

[0035] Velocity continuity constraint: C1(x) t+1 ,x t )=‖(v t+1 -v t )-a t ·Δt‖ 2 Where Δt = 0.1 seconds, x t+1 x t Let v represent the state vectors at time t+1 and time t, respectively. t+1 v t Let a represent the velocities at time t+1 and time t, respectively. t Represents the acceleration at time t;

[0036] Acceleration range constraint: C2(x) t )=max(0,‖a t ||-9g) 2 Where g = 9.8 m / s 2 9g represents the maximum acceleration limit of the aircraft;

[0037] Turning rate constraint: C3(x) t )=max(0,‖ω t ||-25deg / s) 2 25 deg / s is the aircraft's maximum turn rate, ω t Indicates the turning rate;

[0038] Energy conservation approximation constraints: Among them W ext Indicates external work done; m represents the mass of the aircraft, h t+1 h represents the altitude at time t+1. t Indicates the altitude at time t;

[0039] Step 3-2: Physical constraint loss function: Define the physical constraint loss function L phys :

[0040]

[0041] Among them, the coefficients 0.35, 0.25, 0.25 and 0.15 are the weights of each physical constraint, which are determined according to the importance of the constraint;

[0042] Step 3-3: Physical Constraint Optimization: Combining the total loss function and the physical constraint loss function, a new total loss function is obtained:

[0043]

[0044] Where 0.2 is the weighting coefficient for the physical constraint loss;

[0045] The deep Koopman model was fine-tuned using a new total loss function. The number of training epochs was set to 50, and the learning rate was set to 1 / 10 of the original, i.e., 0.0001, resulting in the physically constrained optimized deep Koopman model M. phys ;

[0046] Steps 3-4: Feature Analysis and Dimensionality Reduction: Principal Component Analysis (PCA) and sparse autoencoders are used to analyze the importance of each dimension in the Koopman feature space and select key features to simplify the model structure.

[0047] Principal component analysis: PCA decomposition is performed on the Koopman feature space, and the contribution rate of each principal component is calculated;

[0048] Sparsity analysis: Calculate the L1 norm of each feature dimension and evaluate its sparsity;

[0049] Feature selection: Features corresponding to the principal components that explain 95% of the variance are retained to form a simplified model.

[0050] Preferably, step 4 specifically comprises:

[0051] Step 4-1: Recursive prediction baseline: based on the trained Koupman matrix K ω Perform recursive prediction:

[0052]

[0053] Where h represents the prediction step size; Let represent the predicted state vectors at steps t+1, t+2, and t+h, respectively.

[0054] Step 4-2: Error Analysis: Establish a model relating prediction step size to error, and quantify the impact of prediction duration on accuracy.

[0055]

[0056] Among them, E step (h) represents the average error of the prediction over h steps. Let x represent the predicted state vector at step i+h. i+h This represents the state vector at step i+h;

[0057] Step 4-3: Adaptive Prediction Interval: Based on the error accumulation model, dynamically adjust the prediction interval and introduce multiple prediction starting points:

[0058] Short-term forecast: Using the current time t as the starting point, forecast the interval [t+1, t+10].

[0059] Interim forecast: Using time t-5 as the starting point, predict the interval [t+11, t+20];

[0060] Long-term prediction: Using time t-10 as the starting point, predict the interval [t+21, t+30];

[0061] Step 4-4: Integrated Prediction Fusion: Fusion of multiple prediction results using the Bayesian model averaging method:

[0062]

[0063] in, w represents the prediction result of the l-th model for time t+j. l (j) represents the corresponding weight, satisfying And w l (j)≥0; This represents the predicted state vector of the final fusion;

[0064] weight w l (j) Determine by minimizing the prediction error on the validation set, and solve the following optimization problem using the Lagrange multiplier method:

[0065]

[0066] And w l (j)≥0

[0067] Among them, w * N represents the optimal weight.val Indicates the number of samples in the validation set. Let x represent the predicted state vector of the l-th model. i+j Let w represent the true state vector and w represent the weight vector.

[0068] The quadratic programming algorithm is used in the solution process;

[0069] Steps 4-5: Variational Inference: The uncertainty of the prediction is quantified using the Monte Carlo sampling variational algorithm, and a confidence interval is generated for each prediction point:

[0070]

[0071] Where, σ t+j This represents the standard deviation of the forecast.

[0072] Preferably, step 5 specifically comprises:

[0073] Step 5-1: Change Point Detection: Design a change point detection algorithm to identify changes in aircraft maneuvering patterns.

[0074]

[0075] in, σ represents the prediction error. e The standard deviation of the prediction error. For indicator functions, CP t This represents the detection result of the change point at time t. This represents the predicted state value at time t; when the condition is met for three consecutive time steps, it is confirmed as a point of change.

[0076] Step 5-2: Incremental learning strategy: Adjust the incremental learning strategy based on the detected changes;

[0077] If a change point is detected, model update strategy 1 is adopted: retrain all 3 layers of the encoder network and adjust all parameters of the Koopman matrix, with the learning rate set to 1e-3;

[0078] If no change is detected, adopt model update strategy 2: only adjust the parameters of the last layer of the encoder network and the diagonal elements of the Koopman matrix, and set the learning rate to 5e-5;

[0079] Step 5-3: Sliding window training: The sliding window method is used to train the model by combining new and old data;

[0080]

[0081] in, This represents a representative sample selected from historical data. The specific selection method is as follows: 20% of the samples are randomly selected from the historical database, and then k-means clustering is used to select the 5 samples closest to each cluster center, resulting in a total of 50 representative samples; D new D represents the new trajectory data acquired in real time. window This represents the complete dataset used for sliding window training;

[0082] Step 5-4: Knowledge Transfer: Using model distillation, the predictive power of the old model is transferred to the new model; the old model refers to the currently running, pre-trained deep Koopman model, and the new model refers to the model based on the newly acquired trajectory data D in real time. new The re-parameterized deep Koopman model;

[0083]

[0084] Where, Φ new (.) and Φ old (.) represent the encoders of the new model and the old model, respectively. Ψ new (.) and Ψ old (.) represent the decoders for the new model and the old model, respectively. i For randomly sampled Koopman space points; L transfer Represents the knowledge transfer loss function;

[0085] Knowledge transfer from the old model to the new model:

[0086]

[0087] Among them, f new (.) and f old (.) represent the prediction functions of the new model and the old model, respectively;

[0088] Step 5-5: Model Integration and Update: Integrate the new model with the old model to form the updated model.

[0089] M updated =μM new +(1-μ)M old

[0090] Where μ is the update rate, initially set to 0.3, and dynamically adjusted according to the performance of the new model on the validation set: if the root mean square error of the new model on the validation set is reduced by more than 15% compared to the old model, then μ increases by 0.2; if the root mean square error of the new model on the validation set is improved by less than 5% compared to the old model, then μ decreases by 0.1; if the root mean square error of the new model on the validation set is larger than that of the old model, then μ is set to 0.05.

[0091] Preferably, the structure of the encoder network is as follows:

[0092] Input layer: d nodes, corresponding to the original state vector;

[0093] Hidden layer 1: 128 nodes, activation function is ReLU;

[0094] Hidden layer 2: 64 nodes, activation function is ReLU;

[0095] Output layer: k = 32 nodes, activation function is tanh.

[0096] Preferably, the structure of the decoder network is as follows:

[0097] Input layer: k = 32 nodes, corresponding to Koopman feature vectors;

[0098] Hidden layer 1: 64 nodes, activation function is ReLU;

[0099] Hidden layer 2: 128 nodes, activation function is ReLU;

[0100] Output layer: d nodes, linear activation function.

[0101] Preferably, in steps 2-4, when training the total loss function, the initial learning rate is set to 0.001, and it decays to 0.8 times the original rate every 50 epochs, with the batch size set to 64.

[0102] The beneficial effects of the present invention are as follows:

[0103] This invention combines deep learning with Koopman operator theory to construct a three-layer network that automatically extracts Koopman features. It introduces four physical constraints to improve the model's generalization ability and realizes a three-stage prediction strategy and dual-channel online learning. This solves the problems of low prediction accuracy, poor physical consistency, and weak adaptability of traditional methods in high-maneuverability flight scenarios. It achieves high-precision aircraft trajectory prediction, provides reliable target information for the seeker system, and significantly improves the interception success rate. Attached Figure Description

[0104] Figure 1 This is a comparison chart of the aircraft trajectory prediction effects of embodiments of the present invention. Detailed Implementation

[0105] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0106] This invention proposes an aircraft trajectory prediction method based on the deep learning Koopman operator. This method uses a deep neural network to automatically learn the Koopman feature mapping function, converts the nonlinear flight dynamics system into a high-dimensional linear representation, and combines physical constraints to optimize the prediction model, thereby achieving high-precision, long-term aircraft trajectory prediction.

[0107] Step 1: Trajectory data acquisition and preprocessing;

[0108] This step involves collecting historical trajectory data of the target aircraft using various sensors such as infrared and radar, and then preprocessing the data.

[0109] Step 1-1: Data Acquisition: Obtain the position, velocity, and acceleration status information of the target aircraft through multi-sensor fusion technology to form the original trajectory sequence X. raw ={x1,x2,...,x T},in Let d represent the d-dimensional state vector at time t, where t = 1…t;

[0110] Step 1-2: Noise Filtering: The Unscented Kalman Filter (UKF) method is used to filter noise from the original data. During the filtering process, the state covariance matrix Q is set as a diagonal matrix with diagonal elements [0.01, 0.01, 0.01, 0.05, 0.05, 0.05, 0.1, 0.1, 0.1]. The observation noise covariance matrix R is also set as a diagonal matrix with diagonal elements [0.1, 0.1, 0.1]. The filtered dataset X is obtained. filtered ;

[0111] Steps 1-3: Data Completion: To address data loss caused by sensor occlusion or malfunction, a cubic spline interpolation algorithm is used for data completion. The spline smoothing parameter is set to 0.8 to generate a complete trajectory sequence S. complete ;

[0112] Steps 1-4: Feature Extraction: Extracting position coordinates (po) from the complete trajectory x ,po y ,po z ), velocity (v) x ,v y ,v z ) and acceleration (a x ,a y ,a z Construct a feature vector set S feature ;

[0113] Steps 1-5: Data Standardization: Standardize the feature data to eliminate the influence of differences in the units of different features, resulting in a standardized dataset. where X i ={x i ,x i+1 ,...,x i+p-1 Let} be an input sequence of length p, and Y i ={x i+p ,x i+p+1 ,...,x i+p+q-1Let} be the predicted target sequence of length q;

[0114] Step 2: Construction of the deep Koopman encoder network;

[0115] This step designs a deep neural network architecture that automatically learns the mapping from a nonlinear system to the Koopman feature space.

[0116] Step 2-1: Encoder Network Design: Constructing the Koopman encoder network Φ θ : Where θ represents the network parameters, d is the original state space dimension, and k is the Koopman feature space dimension; the encoder network adopts a multilayer perceptron (MLP) structure.

[0117] Input layer: d nodes, corresponding to the original state vector; Hidden layer 1: 128 nodes, activation function is ReLU; Hidden layer 2: 64 nodes, activation function is ReLU; Output layer: k = 32 nodes, activation function is tanh;

[0118] Step 2-2: Decoder Network Design: Constructing the Decoder Network Ψ φ : Where φ represents the network parameters used to map the state of the Koopman feature space back to the original state space;

[0119] Input layer: k = 32 nodes, corresponding to Koopman feature vectors; Hidden layer 1: 64 nodes, activation function is ReLU; Hidden layer 2: 128 nodes, activation function is ReLU; Output layer: d nodes, linear activation function;

[0120] Steps 2-3: Koopman operator approximation: Constructing the Koopman matrix Where ω is a learnable parameter used to linearly advance the system state in the Koopman feature space;

[0121] Steps 2-4: Network Training: Define the total loss function L total It includes the following parts:

[0122] Reconstruction loss:

[0123] Predicted loss:

[0124] Linear consistency loss:

[0125] Regularization loss: L reg =0.001‖θ‖ 2 +0.001‖φ‖ 2 +0.0005‖ω‖ 2 ;

[0126] Total loss function: L total =0.3L recon +0.5L pred +0.2L lin +L reg ;

[0127] Minimize L using the Adam optimizer total The initial learning rate was set to 0.001, decaying to 0.8 times its original value every 50 epochs. The batch size was set to 64. Parameters θ, φ, and ω were updated through backpropagation to obtain the trained encoder network Φ. θ Decoder Network Ψ φ and the Koopman matrix K ω ;

[0128] Step 3: Physically constrained deep Koopman model;

[0129] This step incorporates physics knowledge into the deep Koopman model, enhancing the model's physical interpretability and predictive accuracy.

[0130] Step 3-1: Definition of Flight Dynamics Constraints: Based on the principles of aircraft flight dynamics, define a set of physical constraints C. phys ={C1,C2,C3,C4}, including:

[0131] Velocity continuity constraint: C1(x) t+1 ,x t )=‖(v t+1 -v t )-a t ·Δt‖ 2 , where Δt = 0.1 seconds;

[0132] Acceleration range constraint: C2(x) t )=max(0,‖a t ||-9g) 2 Where g = 9.8 m / s 2 9g represents the typical maximum acceleration limit of the aircraft;

[0133] Turning rate constraint: C3(x) t )=max(0,‖ω t ||-25deg / s) 2 25 deg / s is the typical maximum turn rate of an aircraft;

[0134] Energy conservation approximation constraints: Among them W ext Estimate by thrust and drag;

[0135] Step 3-2: Physical constraint loss function: Define the physical constraint loss function L phys :

[0136]

[0137] Among them, the coefficients 0.35, 0.25, 0.25 and 0.15 are the weights of each physical constraint, which are determined according to the importance of the constraint;

[0138] Step 3-3: Physical Constraint Optimization: Combining the total loss function and the physical constraint loss function, a new total loss function is obtained:

[0139]

[0140] Where 0.2 is the weighting coefficient for the physical constraint loss;

[0141] The deep Koopman model was fine-tuned using a new total loss function. The number of training epochs was set to 50, and the learning rate was set to 1 / 10 of the original, i.e., 0.0001, resulting in the physically constrained optimized deep Koopman model M. phys ;

[0142] Steps 3-4: Feature Analysis and Dimensionality Reduction: Principal Component Analysis (PCA) and sparse autoencoders are used to analyze the importance of each dimension in the Koopman feature space and select key features to simplify the model structure.

[0143] Principal component analysis: PCA decomposition is performed on the Koopman feature space, and the contribution rate of each principal component is calculated;

[0144] Sparsity analysis: Calculate the L1 norm of each feature dimension and evaluate its sparsity;

[0145] Feature selection: Features corresponding to the principal components that explain 95% of the variance are retained to form a simplified model;

[0146] Step 4: Adaptive multi-step prediction strategy;

[0147] This step designs an adaptive multi-step prediction strategy to address the problem of error accumulation in long-term prediction.

[0148] Step 4-1: Recursive prediction baseline: based on the trained Koupman matrix K ω Perform recursive prediction:

[0149]

[0150] Where h represents the prediction step size, and the basic prediction step size is set to 30 steps (equivalent to 3 seconds, with a sampling rate of 10Hz);

[0151] Step 4-2: Error Analysis: Establish a model relating prediction step size to error, and quantify the impact of prediction duration on accuracy.

[0152]

[0153] Experiments on the training set yielded an error growth curve with step size. Actual measurement results showed that the prediction error grew slowly within h=10 steps, grew at a moderate speed between h=10 and 20 steps, and grew rapidly after h>20 steps.

[0154] Step 4-3: Adaptive Prediction Interval: Based on the error accumulation model, dynamically adjust the prediction interval and introduce multiple prediction starting points:

[0155] Short-term forecast: Using the current time t as the starting point, forecast the interval [t+1, t+10].

[0156] Interim forecast: Using time t-5 as the starting point, predict the interval [t+11, t+20];

[0157] Long-term prediction: Using time t-10 as the starting point, predict the interval [t+21, t+30];

[0158] Step 4-4: Integrated Prediction Fusion: Fusion of multiple prediction results using the Bayesian model averaging method:

[0159]

[0160] in, w represents the prediction result of the l-th model for time t+j. l (j) represents the corresponding weight, satisfying And w l (j)≥0;

[0161] weight w l (j) Determine by minimizing the prediction error on the validation set, and solve the following optimization problem using the Lagrange multiplier method:

[0162]

[0163] And w l (j)≥0

[0164] The quadratic programming algorithm is used in the solution process; the upper limit of the number of iterations is set to 100, and the convergence threshold is set to 1e-6.

[0165] Steps 4-5: Variational Inference: The uncertainty of the prediction is quantified using the Monte Carlo sampling variational algorithm, and a confidence interval is generated for each prediction point:

[0166]

[0167] Where, σ t+j This represents the predicted standard deviation, calculated using the encoder network's forward propagation. A Dropout layer (probability 0.1) is activated for 100 forward samples, and the output standard deviation is calculated. 1.96 represents the z-score corresponding to the 95% confidence interval.

[0168] Step 5: Online learning and model updates;

[0169] This step enables the model to learn and continuously optimize online, adapting to changes in the target aircraft's maneuverability.

[0170] Step 5-1: Change Point Detection: Design a change point detection algorithm to identify significant changes in aircraft maneuvering patterns.

[0171]

[0172] in, σ represents the prediction error. e The standard deviation of the prediction error is calculated over the most recent 30 time steps; This is an indicator function; when the condition is met for three consecutive time steps, it is identified as a point of change.

[0173] Step 5-2: Incremental learning strategy: Adjust the incremental learning strategy based on the detected changes;

[0174] If a change point is detected, model update strategy 1 is adopted: retrain all 3 layers of the encoder network and adjust all parameters of the Koopman matrix, with the learning rate set to 1e-3;

[0175] If no change is detected, adopt model update strategy 2: only adjust the parameters of the last layer of the encoder network and the diagonal elements of the Koopman matrix, and set the learning rate to 5e-5;

[0176] Step 5-3: Sliding window training: The sliding window method is used to train the model by combining new and old data;

[0177]

[0178] in, This represents the representative samples selected from historical data. The specific selection method is as follows: 20% of the samples are randomly selected from the historical database, and then the 5 closest samples to each cluster center are selected through k-means clustering, for a total of 50 representative samples.

[0179] Step 5-4: Knowledge Transfer: Transfer the predictive power of the old model to the new model through model distillation techniques;

[0180]

[0181] Where, Φ new (.) and Φ old (.) represent the encoders of the new model and the old model, respectively. Ψ new (.) and Ψ old (.) represent the decoders for the new model and the old model, respectively. i The points are randomly sampled in Koopman space; training uses a temperature parameter T = 2.0, a learning rate of 0.0005, and 30 training epochs; L transfer This represents the knowledge transfer loss function, used to ensure that the new model can inherit the effective predictive capabilities of the old model;

[0182] Knowledge transfer from the old model to the new model:

[0183]

[0184] Among them, f new (.) and f old (.) represent the prediction functions of the new model and the old model, respectively;

[0185] Step 5-5: Model Integration and Update: Integrate the new model with the old model to form the updated model.

[0186] M updated =μM new +(1-μ)M old

[0187] Where μ is the update rate, initially set to 0.3, and dynamically adjusted based on the performance of the new model on the validation set; the root mean square error (RMSE) is used as the core metric for performance evaluation, and the calculation formula is as follows:

[0188]

[0189] The validation set contains 300 trajectory segments with different maneuvering patterns, each segment being 30 time steps long. The value of μ depends on the change in the root mean square error (RMSE) of the new model on the validation set compared to the old model. If the RMSE of the new model on the validation set is reduced by more than 15% compared to the old model, μ increases by 0.2; if the RMSE of the new model on the validation set is reduced by less than 5% compared to the old model, μ decreases by 0.1; if the RMSE of the new model on the validation set is larger than that of the old model, μ is set to 0.05.

[0190] Step 6: System integration and application deployment;

[0191] This step integrates the trained model into the seeker system to achieve real-time trajectory prediction.

[0192] Step 6-1: Model Compression: Structured pruning and quantization techniques are used to compress the model size and reduce computational complexity.

[0193] Weight pruning: Use L1 regularization to sparsify the weights and remove weight parameters whose absolute value is less than the threshold of 0.01;

[0194] Quantization: A linear quantization algorithm is used to quantize the 32-bit floating-point weights into 8-bit integers, with a quantization ratio of (max-min) / 255;

[0195] Structural optimization: The hidden layers of the encoder and decoder were reduced from [128,64] and [64,128] to [64,32] and [32,64], respectively;

[0196] Step 6-2: Hardware Adaptation: Model optimization is performed on the seeker DSP chip (TI TMS320C6678) to achieve real-time calculation.

[0197] Instruction set optimization: Accelerate matrix operations using DSP's SIMD instructions;

[0198] Memory layout: Optimize data storage structure to reduce cache misses;

[0199] Parallel computing: Utilizes a multi-core architecture to achieve parallel processing with 2 threads, keeping the computation latency within 10ms;

[0200] Step 6-3: System Integration: Integrate the trajectory prediction module with the guidance law module and the control system to form a complete interception system.

[0201] Prediction interface design: The input is the aircraft state data of the most recent 20 time steps, and the output is the predicted trajectory for the next 30 time steps;

[0202] Data transmission protocol: UDP protocol, transmission frequency 100Hz, delay tolerance 20ms;

[0203] Guidance law integration: The predicted trajectory is directly input into the proportional guidance law algorithm, with a guidance gain coefficient of 3.5.

[0204] Step 6-4: Performance Monitoring: Design a threshold-based real-time monitoring system to monitor key indicators.

[0205] Error monitoring: Calculates short-term (1s) prediction error per second, with a threshold of 10m;

[0206] Delay monitoring: Measure the prediction calculation time and issue a warning if it exceeds 15ms;

[0207] Data logging: Performance metrics are recorded every minute, including average error, maximum error, and computation latency;

[0208] Step 6-5: Fault Tolerance Mechanism: Implement a multi-level fault protection mechanism to ensure system reliability.

[0209] Data anomaly handling: When an anomaly is detected, use valid data from the previous time step to replace it;

[0210] Resource shortage handling: When computational resources are insufficient, switch to a lightweight model (halve the prediction step size);

[0211] Backup system: Maintain an unscented Kalman filter as a backup, which automatically switches over when the main system fails;

[0212] like Figure 1 As shown, to verify the effectiveness of the method of the present invention, a dataset of aircraft trajectories in a real environment was collected, and a comparative experiment was conducted with existing mainstream trajectory prediction methods. The experimental results are shown in Table 1:

[0213] Table 1

[0214]

[0215] As shown in the table, the method of this invention significantly outperforms existing methods in short-term, medium-term, and long-term prediction accuracy. Particularly in long-term prediction (5 seconds), the error is reduced by 33.2% compared to the second-best method (GRU-physics hybrid model). Simultaneously, the method of this invention maintains a high physical consistency score, indicating that the prediction results conform to flight dynamics. In terms of computation time, although slightly longer than the Kalman filter method, it still meets the requirements for real-time processing and its computational efficiency is superior to other deep learning methods.

[0216] The innovative aspects of this invention:

[0217] 1. Deep Koopman Coding Network: This invention automatically learns the Koopman feature representation of complex nonlinear systems through a three-layer neural network architecture ([d→128→64→32]), replacing the manually designed observation function in the traditional Koopman method. The feature representation capability is improved by 89%, enabling the Koopman method to be successfully applied to the problem of predicting high-dimensional complex flight trajectories.

[0218] 2. Integration of Flight Dynamics Constraints: This invention integrates flight dynamics knowledge into a deep learning framework in the form of explicit mathematical constraints, including velocity continuity constraints, 9g acceleration limit constraints, 25 degrees / second turn rate constraints, and energy conservation constraints. This enables the model to maintain both data-driven adaptability and generalization ability based on a physics model, reducing errors by 37.5%.

[0219] 3. Adaptive Multi-Segment Prediction Strategy: This invention designs a segmented prediction and fusion strategy, dividing the 30-step prediction into 3 intervals: short-term [t+1, t+10], medium-term [t+11, t+20], and long-term [t+21, t+30]. The results are fused using a Bayesian model averaging method, reducing the long-term prediction error by 33.2%.

[0220] 4. Dual-channel update mechanism: This invention implements a dual-channel online learning mechanism based on change point detection. The change point detection threshold is 2.5σ. When there is a change, all three layers of network parameters are retrained (learning rate 1e-3). When there is no change, only the parameters of the last layer are adjusted (learning rate 5e-5), which improves the model's adaptability by 42%.

[0221] In summary, this invention combines deep learning with Koopman operator theory to construct a three-layer network for automatic extraction of Koopman features, introduces four physical constraints to improve the model's generalization ability, and implements a three-stage prediction strategy and dual-channel online learning. This solves the problems of low prediction accuracy, poor physical consistency, and weak adaptability of traditional methods in high-maneuverability flight scenarios, achieving high-precision aircraft trajectory prediction, providing reliable target information for the seeker system, and significantly improving the interception success rate.

Claims

1. A method for predicting aircraft trajectory based on the deep Koopman operator, characterized in that, Includes the following steps: Step 1: Trajectory data acquisition and preprocessing; Step 2: Deep Koopman model construction; Step 2-1: Encoder Network Design: Constructing the encoder network Φ θ : Where θ represents the network parameters, d is the original state space dimension, and k is the Koopman feature space dimension; The encoder network adopts a multilayer perceptron (MLP) structure; Step 2-2: Decoder Network Design: Constructing the Decoder Network Ψ φ : Where φ represents the network parameters used to map the state of the Koopman feature space back to the original state space; Steps 2-3: Koopman operator approximation: Constructing the Koopman matrix Where ω is a learnable parameter used to linearly advance the system state in the Koopman feature space; Steps 2-4: Network Training: Define the total loss function L total It includes the following parts: Reconstruction loss: Predicted loss: Linear consistency loss: Regularization loss: L reg =0.001‖θ‖ 2 +0.001‖φ‖ 2 +0.0005‖ω‖ 2 ; Total loss function: L total =0.3L recon +0.5L pred +0.2L lin +L reg ; Minimize L using the Adam optimizer total The trained encoder network Φ is obtained by updating the parameters θ, φ, and ω through backpropagation. θ Decoder Network Ψ φ and the Koopman matrix K ω ; In the formula, N represents the total number of samples in the dataset, x i x i+1 To input data, x i+p The output data is p, where p is the length of the input sequence. Step 3: Physically constrained deep Koopman model; Based on the principles of aircraft flight dynamics, a set of physical constraints is defined, including: Velocity continuity constraint; Acceleration range constraints; Turning rate constraint; Energy conservation approximation constraint; Step 4: Adaptive multi-step prediction strategy; Recursive prediction is performed based on the trained Koppman matrix; Establish a model to show the relationship between prediction step size and error, and quantify the impact of prediction duration on accuracy; Based on the model of the relationship between prediction step size and error, the prediction interval is dynamically adjusted, and multiple prediction starting points are introduced. Multiple prediction results are fused using the Bayesian model averaging method. The uncertainty of the prediction is quantified using the Monte Carlo sampling variational algorithm, and a confidence interval is generated for each prediction point; Step 5: Online learning and model updates.

2. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 1, characterized in that, Step 1 specifically involves: Step 1-1: Data Acquisition: Obtain the position, velocity, and acceleration status information of the target aircraft through multi-sensor fusion technology to form the original trajectory sequence X. raw ={x1,x2,...,x T },in Let d represent the d-dimensional state vector at time t, where t = 1…T; Step 1-2: Noise Filtering: The Unscented Kalman Filter (UKF) method is used to filter noise from the original data. During the filtering process, the state covariance matrix Q is set as a diagonal matrix with diagonal elements [0.01, 0.01, 0.01, 0.05, 0.05, 0.05, 0.1, 0.1, 0.1]. The observation noise covariance matrix R is also set as a diagonal matrix with diagonal elements [0.1, 0.1, 0.1]. The filtered dataset X is obtained. filtered ; Steps 1-3: Data Completion: To address data loss caused by sensor occlusion or malfunction, a cubic spline interpolation algorithm is used for data completion. The spline smoothing parameter is set to 0.8 to generate a complete trajectory sequence S. complete ; Steps 1-4: Feature Extraction: Extracting position coordinates (po) from the complete trajectory x ,po y ,po z ), velocity (v) x ,v y ,v z ) and acceleration (a x ,a y ,a z Construct a feature vector set S feature ; Steps 1-5: Data Standardization: Standardize the feature data to eliminate the influence of differences in the units of different features, resulting in a standardized dataset. where X i ={x i ,x i+1 ,...,x i+p-1 Let} be an input sequence of length p, and Y i ={x i+p ,x i+p+1 ,...,x i+p+q-1 Let} be the target sequence to be predicted, which is of length q.

3. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 2, characterized in that, Step 3 specifically involves: Step 3-1: Definition of Flight Dynamics Constraints: Based on the principles of aircraft flight dynamics, define a set of physical constraints C. phys ={C1,C2,C3,C4}, including: Velocity continuity constraint: C1(x) t+1 ,x t )=‖(v t+1 -v t )-a t ·Δt‖ 2 Where Δt = 0.1 seconds, x t+1 x t Let v represent the state vectors at time t+1 and time t, respectively. t+1 v t Let a represent the velocities at time t+1 and time t, respectively. t Represents the acceleration at time t; Acceleration range constraint: C2(x) t )=max(0,‖a t ||-9g) 2 Where g = 9.8 m / s 2 9g represents the maximum acceleration limit of the aircraft; Turning rate constraint: C3(x) t )=max(0,‖ω t ||-25deg / s) 2 25 deg / s is the aircraft's maximum turn rate, ω t Indicates the turning rate; Energy conservation approximation constraints: Among them W ext Indicates external work done; m represents the mass of the aircraft, h t+1 h represents the altitude at time t+1. t Indicates the altitude at time t; Step 3-2: Physical constraint loss function: Define the physical constraint loss function L phys : Among them, the coefficients 0.35, 0.25, 0.25 and 0.15 are the weights of each physical constraint, which are determined according to the importance of the constraint; Step 3-3: Physical Constraint Optimization: Combining the total loss function and the physical constraint loss function, a new total loss function is obtained: Where 0.2 is the weighting coefficient for the physical constraint loss; The deep Koopman model was fine-tuned using a new total loss function. The number of training epochs was set to 50, and the learning rate was set to 1 / 10 of the original, i.e., 0.0001, resulting in the physically constrained optimized deep Koopman model M. phys ; Steps 3-4: Feature Analysis and Dimensionality Reduction: Principal Component Analysis (PCA) and sparse autoencoders are used to analyze the importance of each dimension in the Koopman feature space and select key features to simplify the model structure. Principal component analysis: PCA decomposition is performed on the Koopman feature space, and the contribution rate of each principal component is calculated; Sparsity analysis: Calculate the L1 norm of each feature dimension and evaluate its sparsity; Feature selection: Features corresponding to the principal components that explain 95% of the variance are retained to form a simplified model.

4. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 3, characterized in that, Step 4 specifically involves: Step 4-1: Recursive prediction baseline: based on the trained Koupman matrix K ω Perform recursive prediction: Where h represents the prediction step size; Let represent the predicted state vectors at steps t+1, t+2, and t+h, respectively. Step 4-2: Error Analysis: Establish a model relating prediction step size to error, and quantify the impact of prediction duration on accuracy. Among them, E step (h) represents the average error of the prediction over h steps. Let x represent the predicted state vector at step i+h. i+h This represents the state vector at step i+h; Step 4-3: Adaptive Prediction Interval: Based on the relationship model between prediction step size and error, dynamically adjust the prediction interval and introduce multiple prediction starting points: Short-term forecast: Using the current time t as the starting point, forecast the interval [t+1, t+10]. Interim forecast: Using time t-5 as the starting point, predict the interval [t+11, t+20]; Long-term prediction: Using time t-10 as the starting point, predict the interval [t+21, t+30]; Step 4-4: Integrated Prediction Fusion: Fusion of multiple prediction results using the Bayesian model averaging method: in, w represents the prediction result of the l-th model for time t+j. l (j) represents the corresponding weight, satisfying And w l (j)≥0; This represents the predicted state vector of the final fusion; weight w l (j) Determine by minimizing the prediction error on the validation set, and solve the following optimization problem using the Lagrange multiplier method: And w l (j)≥0; Among them, w * N represents the optimal weight. val Indicates the number of samples in the validation set. Let x represent the predicted state vector of the l-th model. i+j Let w represent the true state vector and w represent the weight vector. The quadratic programming algorithm is used in the solution process; Steps 4-5: Variational Inference: The uncertainty of the prediction is quantified using the Monte Carlo sampling variational algorithm, and a confidence interval is generated for each prediction point: Where, σ t+j This represents the standard deviation of the forecast.

5. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 4, characterized in that, Step 5 specifically involves: Step 5-1: Change Point Detection: Design a change point detection algorithm to identify changes in aircraft maneuvering patterns. in, σ represents the prediction error. e The standard deviation of the prediction error. For indicator functions, CP t This represents the detection result of the change point at time t. This represents the predicted state value at time t; when the condition is met for three consecutive time steps, it is confirmed as a point of change. Step 5-2: Incremental learning strategy: Adjust the incremental learning strategy based on the detected changes; If a change point is detected, model update strategy 1 is adopted: retrain all 3 layers of the encoder network and adjust all parameters of the Koopman matrix, with the learning rate set to 1e-3; If no change is detected, adopt model update strategy 2: only adjust the parameters of the last layer of the encoder network and the diagonal elements of the Koopman matrix, and set the learning rate to 5e-5; Step 5-3: Sliding window training: The sliding window method is used to train the model by combining new and old data; in, This represents a representative sample selected from historical data. The specific selection method is as follows: 20% of the samples are randomly selected from the historical database, and then k-means clustering is used to select the 5 samples closest to each cluster center, resulting in a total of 50 representative samples; D new D represents the new trajectory data acquired in real time. window This represents the complete dataset used for sliding window training; Step 5-4: Knowledge Transfer: Using model distillation, the predictive power of the old model is transferred to the new model; the old model refers to the currently running, pre-trained deep Koopman model, and the new model refers to the model based on the newly acquired trajectory data D in real time. new The re-parameterized deep Koopman model; Where, Φ new (.) and Φ old (.) represent the encoders of the new model and the old model, respectively. Ψ new (.) and Ψ old (.) represent the decoders for the new model and the old model, respectively. i For randomly sampled Koopman space points; L transfer Represents the knowledge transfer loss function; Knowledge transfer from the old model to the new model: Among them, f new (.) and f old (.) represent the prediction functions of the new model and the old model, respectively; Step 5-5: Model Integration and Update: Integrate the new model with the old model to form the updated model. M updated =μM new +(1-μ)M old ; Where μ is the update rate, initially set to 0.3, and dynamically adjusted according to the performance of the new model on the validation set: if the root mean square error of the new model on the validation set is reduced by more than 15% compared to the old model, then μ increases by 0.2; if the root mean square error of the new model on the validation set is improved by less than 5% compared to the old model, then μ decreases by 0.1; if the root mean square error of the new model on the validation set is larger than that of the old model, then μ is set to 0.

05.

6. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 5, characterized in that, The structure of the encoder network is as follows: Input layer: d nodes, corresponding to the original state vector; Hidden layer 1: 128 nodes, activation function is ReLU; Hidden layer 2: 64 nodes, activation function is ReLU; Output layer: k = 32 nodes, activation function is tanh.

7. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 6, characterized in that, The structure of the decoder network is as follows: Input layer: k = 32 nodes, corresponding to Koopman feature vectors; Hidden layer 1: 64 nodes, activation function is ReLU; Hidden layer 2: 128 nodes, activation function is ReLU; Output layer: d nodes, linear activation function.

8. The aircraft trajectory prediction method based on the deep Koopman operator according to claim 7, characterized in that, In steps 2-4, when training the total loss function, the initial learning rate is set to 0.001, and it decays to 0.8 times the original rate every 50 epochs. The batch size is set to 64.

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