Anti-error data injection Kalman filtering method based on neural network enhancement

Abstract

The invention relates to the technical field of intelligent control and information security crossing, in particular to an anti-error data injection Kalman filtering method based on neural network enhancement, which comprises the following steps: firstly, constructing a neural network auxiliary correction module for identifying and correcting an extended Kalman filtering intermediate variable affected by an error data injection attack; secondly, designing a hybrid filtering architecture, and combining the learning ability of a neural network with the theoretical advantages of traditional Kalman filtering; and finally, realizing adaptive optimization of model parameters through a mixed training mechanism. Accurate state estimation can be achieved only through limited prior information, and meanwhile the calculation efficiency and interpretability of a traditional Kalman filter are kept. And the anti-interference capability of the system in a non-Gaussian noise environment is effectively improved. The application of the method on an induction motor model is obviously better than that of a traditional anti-error data injection scheme, and breakthrough progress is achieved in the aspects of estimation precision and robustness.

Description

Technical Field

[0001] The present invention relates to the technical field of intersection of intelligent control and information security, and in particular to a Kalman filtering method based on neural network enhancement and resistant to error data injection. Background Art

[0002] State estimation is a core technology for dynamic system monitoring and control. Its goal is to infer the hidden internal state of a system in real time from sensor measurements. The Kalman filter (KF) is widely used for state estimation in linear systems due to its optimality and efficiency in noisy environments. For nonlinear systems, derivative methods such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) have expanded the application scope of the KF through linearization or sampling strategies. However, the performance of these methods is severely degraded when faced with false data injection attacks (FDIA). By tampering with sensor data (e.g., additive attacks and multiplicative attacks), attackers can mislead the state estimation results and thus undermine system stability.

[0003] Existing strategies for addressing FDIA primarily fall into two categories: model-driven and data-driven. Model-driven approaches typically design robust filtering algorithms based on attack statistical models. However, these approaches require precise prior information such as the attack type and noise distribution, and struggle to handle complex attack scenarios in non-Gaussian noise environments. Data-driven approaches employ neural networks to directly learn state mapping relationships from data. While these approaches avoid model assumptions, they suffer from inherent drawbacks such as reliance on massive amounts of training data and poor model interpretability. Hybrid approaches have emerged in recent years, attempting to combine the advantages of state-space models and neural networks, such as using neural networks to replace traditional Kalman gain calculations. However, existing hybrid approaches still suffer from several key technical bottlenecks: they fail to specifically correct for intermediate variables directly impacted by FDIA and are only applicable in Gaussian noise environments; they perform poorly in partially observable systems where the measurement dimension is smaller than the state dimension; and their performance often degrades significantly in heavy-tailed noise environments, which are common in practical engineering applications.

[0004] Therefore, there is an urgent need for a state estimation method that can simultaneously resist multiple types of FDIA (additive and multiplicative attacks) and adapt to non-Gaussian noise under limited prior information. Summary of the Invention

[0005] The purpose of the present invention is to provide a neural network-enhanced Kalman filtering method for resisting error data injection, so as to solve the problem that the performance of traditional Kalman filters is significantly degraded when attacked by error data injection.

[0006] To achieve the above object, the present invention provides a Kalman filtering method for resisting error data injection based on neural network enhancement, comprising the following steps:

[0007] Two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture;

[0008] The dual neural network correction architecture is embedded in the EKF framework, and the posterior state estimate is updated through the neural network output of the corrected residual and the inverse of the covariance matrix, integrating the hybrid filter model;

[0009] Two loss functions are designed according to the type of available dataset, and an alternating training strategy is used to optimize the neural network parameters.

[0010] Among them, two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture, in which:

[0011] A gated recurrent unit network is used to replace the observation residual calculation in the traditional EKF. The specific network structure has three layers, namely a fully connected input layer, a GRU time series modeling layer and a fully connected output layer. The input layer is a 224-neuron fully connected layer, which receives three types of feature inputs: time series difference of the attacked measurement value, innovation difference and predicted measurement time series change.

[0012] The time series difference of the attacked measurement value is

[0013] The innovation gap is

[0014] The predicted measurement time series change is

[0015] Among them, z k It represents the measurement value affected by the error data injection attack at the kth moment, and its expression is:

[0016]

[0017] Among them, y k is the true measurement value without attack, a k and m k are additive and multiplicative attack data respectively, and is a Bernoulli random variable, representing whether the attack occurs, where 1 indicates it occurs and 0 indicates it does not occur. is the predicted measurement value of EKF, which is predicted by the state Calculated by measuring the function h(·):

[0018]

[0019] The input layer maps the input parameters to a higher dimension by combining a fully connected layer with 224 neurons and a ReLU nonlinear activation function to perform feature space mapping;

[0020] 58-unit unidirectional GRU layer to capture the dynamic temporal dependencies of the measurement data;

[0021] The output layer is a fully connected layer with the same dimension as the state, and uses linear activation to output the residual correction Δy k (θ1), the observation residual correction network dynamically learns the attack interference pattern and outputs the corrected residual Δy k (θ1).

[0022] Among them, two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture, in which:

[0023] Design another GRU network output residual covariance matrix inverse The specific network structure has four layers, namely, fully connected input layer, GRU time series modeling layer, fully connected output layer and positive definite matrix output layer.

[0024] Among them, design another GRU network output residual covariance matrix inverse The specific network structure has four layers, namely, fully connected input layer, GRU time series modeling layer, fully connected output layer and positive definite matrix output layer.

[0025] The input layer is a 560-neuron fully connected layer that receives Δz k and the prior state covariance matrix P of EKF k|k-1 Three types of feature inputs are mapped into feature space through ReLU nonlinear activation function to map the input parameters to a higher dimension;

[0026] 58-unit unidirectional GRU layer to capture the dynamic temporal dependencies of the measurement data;

[0027] The output layer is a fully connected layer with the same dimension as the state, and the non-positive definite matrix A is generated;

[0028] The positive definite matrix output layer is passed Where η is a small positive scalar, I m Represents the identity matrix of dimension m, where m is the observation dimension.

[0029] The dual neural network correction architecture is embedded in the EKF framework. The neural network outputs the corrected residual and the inverse of the covariance matrix to update the posterior state estimate and integrate the hybrid filter model. The specific steps include:

[0030] The prediction step of the EKF is retained, and only the update step is enhanced by the neural network. The posterior state estimate is calculated as follows:

[0031]

[0032] Among them, P k|k (θ2) represents the posterior state covariance matrix, and n is the state dimension.

[0033] Among them, two loss functions are designed according to the type of available dataset, and an alternating training strategy is used to optimize the neural network parameters. The specific steps include:

[0034] When the latent state data is available, the mean square error loss function loss1 is used:

[0035]

[0036] Among them, D1 consists of L trajectories of the true state and the attacked measurement value with a length of T. is represented as the true state value of the kth moment of the lth trajectory, and is the resulting posterior state.

[0037] Among them, two loss functions are designed according to the type of available datasets, and an alternating training strategy is used to optimize the neural network parameters. The specific steps also include:

[0038] When the original observation data is available, the measurement residual loss function loss2 is used:

[0039]

[0040] Among them, D2 consists of L trajectories of unattacked and attacked measurement values ​​of length T, is After the state transition function f(·) and the observation function, h(·) is obtained:

[0041]

[0042] Among them, two loss functions are designed according to the type of available datasets, and an alternating training strategy is used to optimize the neural network parameters. The specific steps also include:

[0043] When updating θ1, the gradient is calculated and freeze θ2;

[0044] When updating θ2, the gradient is calculated And freeze θ1.

[0045] The neural network-enhanced Kalman filtering method for resisting error data injection of the present invention, through the collaborative design of the dual neural network correction architecture and the hybrid training strategy, retains the efficiency and interpretability of the traditional extended Kalman filtering framework while significantly improving the robustness of the state estimation system in complex environments. This method can effectively resist various types of error data injection attacks, including additive attacks, multiplicative attacks and their hybrid modes, ensuring that the system can still maintain accurate state estimation when the sensor data is maliciously tampered with. By directly learning the inverse of the residual covariance matrix through the neural network, it gets rid of the dependence on the statistical characteristics of the noise, making it show excellent adaptability in non-Gaussian environments such as heavy-tailed noise. Compared with traditional methods, the present invention only requires the basic state space model of the system, which greatly reduces the need for prior knowledge such as attack statistical characteristics, and at the same time breaks through the limitations of traditional methods in some observable systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art.

[0047] Figure 1 This is a filtering flow chart after reconstruction through the dual neural network correction architecture of the present invention.

[0048] Figure 2 It is the training convergence graph of the present invention that is not attacked by erroneous data.

[0049] Figure 3 It is the training convergence diagram of the present invention subjected only to additive attack.

[0050] Figure 4 It is the training convergence diagram of the present invention subjected only to multiplicative attack.

[0051] Figure 5 It is a training convergence diagram of the present invention that is subjected to both additive and multiplicative attacks.

[0052] Figure 6 The present invention is a flowchart of the steps of the neural network enhanced anti-error data injection Kalman filtering method. DETAILED DESCRIPTION

[0053] The embodiments of the present invention are described in detail below. Examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, but should not be understood as limiting the present invention.

[0054] The first embodiment of this application is:

[0055] See also Figures 1 to 6 ,in, Figure 1 This is a filtering flow chart after reconstruction through the dual neural network correction architecture of the present invention. Figure 2 It is the training convergence graph of the present invention that is not attacked by erroneous data. Figure 3 It is the training convergence diagram of the present invention subjected only to additive attack. Figure 4 It is the training convergence diagram of the present invention subjected only to multiplicative attack. Figure 5 It is a training convergence diagram of the present invention that is subjected to both additive and multiplicative attacks. Figure 6 The present invention is a flowchart of the steps of the neural network enhanced anti-error data injection Kalman filtering method.

[0056] The present invention provides a Kalman filtering method for resisting error data injection based on neural network enhancement, comprising the following steps:

[0057] S101: Two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture;

[0058] S102: Embed the dual neural network correction architecture into the EKF framework, update the posterior state estimate through the neural network output of the corrected residual and the inverse of the covariance matrix, and integrate the hybrid filter model;

[0059] S103: Design two loss functions based on the available dataset types and use an alternating training strategy to optimize the neural network parameters.

[0060] Specifically, a neural network enhanced Kalman hybrid filter model was constructed, which includes a neural network module and a standard extended Kalman filter module; the neural network module corrects the attacked observation information, and the extended Kalman filter module performs state filtering according to the output of the neural network.

[0061] The neural network enhanced Kalman mixture filtering process is as follows:

[0062] 1. Initialization. Initialize the dual neural network parameters θ1 and θ2, and set the initial state value of EKF and its covariance matrix P 0|0 Determine the system dynamic model (state transition function f(·), observation function h(·), process noise covariance Q);

[0063] 2. State prediction: based on the posterior state estimation of the previous moment and its covariance P k-1|k-1 Calculate the prior state estimate at the current k moment and its covariance P k|k-1 :

[0064]

[0065] in, is the state transition function f(·) in Next, the predicted observation value is calculated based on the predicted prior state estimate:

[0066]

[0067] 3. Calculate network input feature variables: Calculate the network input variables of the current moment observation residual correction network and the inverse covariance matrix estimation network based on the current moment prediction value, the attacked observation value, and the relevant information of the previous moment:

[0068]

[0069] 4. Calculate the dual neural network output: Get the residual Δy corrected by the network at the current moment through the network input variable k (θ1) and the inverse of the residual covariance matrix estimated by the network

[0070]

[0071] 5. State update: Update the posterior state and posterior state covariance matrix based on the prior information obtained from the state prediction and the residual and residual covariance matrix output by the neural network:

[0072]

[0073] The dual neural network update process is as follows:

[0074] 1. Potential state data is available: use the mean square error loss function loss1:

[0075]

[0076] in, It is the state filter value obtained by the neural network enhanced Kalman mixture filtering process. Based on the loss function loss1, the gradient of the dual neural network parameters θ1, θ2 is calculated as follows:

[0077]

[0078]

[0079] in, and The calculation is as follows:

[0080]

[0081] 2. Unattacked observation data is available: the measurement residual loss function loss2 is used:

[0082]

[0083] in, The state filter value at time k-1 obtained by the neural network enhanced Kalman hybrid filter process is obtained through the state transition function f(·) and the observation function h(·). Based on the loss function loss2, the gradient of the dual neural network parameters θ1 and θ2 is calculated as follows:

[0084]

[0085] in, The calculation is as follows:

[0086]

[0087] Simulation experiment analysis:

[0088] To validate the effectiveness of the proposed dual neural network-enhanced Kalman filtering method (FDI-Net), comparative experiments were conducted in an induction motor state estimation scenario, encompassing both Gaussian and non-Gaussian noise environments. The results were tested under four attack scenarios: no attack, additive attack only, multiplicative attack only, and mixed attack. The comparison algorithms included the traditional EKF, the attack-resistant model EKF_FD, and the ideal EKF without attack.

[0089] The experiment uses the mean square error (MSE) in dB as the core evaluation indicator, which is defined as:

[0090] 1. System model: The noiseless discrete fifth-order nonlinear induction motor model is defined as follows:

[0091]

[0092] Among them, u k,1 =350cos(0.003k) and u k,2 =350sin(0.003k) represents the input stator voltage signal.

[0093] Rotor time constant T r and its related parameters σ, and ρ is defined as:

[0094]

[0095] Among them, the parameter R S 、R r , M, L s 、L s , J, T r, P, and λ are constants in the motor model, representing different motor parameters, including resistance, inductance, and rotor time constant.

[0096] 2. Experimental settings: Attack parameters: additive attack a k ~N(5×1 5×4 ,4×1 5×5 ), multiplicative attack m k ~N(0.9×1 5×1 ,0.3×1 5×5 ), the probability of attack occurring p a =p m =0.2.

[0097] Noise configuration: Gaussian noise: R=10 -3 I 2×2 ; Non-Gaussian noise (heavy-tailed noise):

[0098]

[0099] Where W k is the noise at the kth moment, is a Bernoulli random variable.

[0100] Training configuration: The training set contains 500 trajectories with a length of 50 to fully train the network; the test set contains 30 trajectories with a length of 80 to verify the generalization ability of long sequences; the training batch size is set to 16 and the learning rate is set to 0.0006.

[0101] 3. Experimental results:

[0102] 3.1. Gaussian noise environment: the observation noise covariance matrix is ​​R = 10 -3 I 2×2

[0103] No attack scenario: the probability of attack occurring p a =p m =0.2. Figure 2 As shown in Figure 3, the performance of FDI-Net (including the version FDI-Net-S trained with loss1 and the version FDI-Net-O trained with loss2) is comparable to that of the standard EKF, demonstrating its basic filtering capabilities.

[0104] Only additive attack scenario: that is, the probability of attack p a =0.2, p m = 0. Figure 3As shown in the figure, FDI-Net significantly outperforms the attacked EKF and EKF_FD, and even approaches the performance of the unattacked EKF. Because FDI-Net-O uses labeled pairs of attacked and normal observations as its training set, the network, under the direct guidance of normal observations, can more accurately learn the behavioral characteristics of the error data injection attack, thus exhibiting smoother convergence characteristics. In contrast, FDI-Net-S uses labeled pairs of attacked observations and latent states as its training data. Although it can learn the implicit mapping relationship between attacked observations and latent states, due to the presence of noise interference, the network has difficulty effectively distinguishing measurement noise from attack information during training, resulting in slight fluctuations in its convergence process.

[0105] Only multiplicative attack scenario: that is, the probability of attack p a =0, p m =0.2. Figure 4 As shown in Figure 3, FDI-Net significantly outperforms the attacked EKF and EKF_FD.

[0106] Mixed attack scenario: the probability of attack occurring p a =p m =0.2. Figure 5 As shown in Figure 2, FDI-Net significantly outperforms the attacked EKF. Since EKF_FD does not have the ability to handle both additive and multiplicative attacks, we will not compare them here.

[0107] 3.2. Non-Gaussian noise environment: This paper mainly discusses heavy-tailed observation noise, as shown in Table 1: MSE (dB) of various scenarios under heavy-tailed noise.

[0108]

[0109] Table 1

[0110] When there is no attack, the MSE of FDI-Net-S is -8.48dB, which is better than EKF (-3.35dB).

[0111] Under additive attacks only, the MSE of FDI-Net-S (-7.07dB) far exceeds that of the attacked EKF (3.87dB).

[0112] Under hybrid attacks, the MSE of FDI-Net-S (-3.27dB) is significantly improved compared to the attacked EKF (30.29dB).

[0113] Based on the traditional extended Kalman filter framework, this paper innovatively proposes a robust state estimation method FDI-Net that integrates dual neural networks. Compared with existing anti-attack filtering algorithms, this method adaptively corrects observation residuals and covariance matrices in a data-driven manner, significantly reducing dependence on attack prior information. Simulation experiments show that in Gaussian and non-Gaussian (heavy-tailed) noise environments, facing additive, multiplicative, and mixed error data injection attacks, the estimation accuracy of FDI-Net is significantly better than that of traditional EKF and model anti-attack method EKF_FD; especially in partially observable scenarios, its MSE performance is close to the ideal case of no attack, providing a reliable technical solution for security status monitoring of critical infrastructure.

[0114] The above disclosure is merely one or more preferred embodiments of the present application and is not intended to limit the scope of the present application. A person skilled in the art will understand that all or part of the processes of the above embodiments and equivalent changes made in accordance with the claims of the present application are still within the scope of the present application.

Claims

1. A Kalman filter method based on neural network enhancement to resist error data injection, characterized in that: The following steps are involved: Two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture; The dual neural network correction architecture is embedded in the EKF framework, and the posterior state estimate is updated through the neural network output of the corrected residual and the inverse of the covariance matrix, integrating the hybrid filter model; Two loss functions are designed according to the type of available dataset, and an alternating training strategy is used to optimize the neural network parameters.

2. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 1, wherein: Two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture, in which: A gated recurrent unit network is used to replace the observation residual calculation in the traditional EKF. The specific network structure has three layers, namely a fully connected input layer, a GRU time series modeling layer and a fully connected output layer. The input layer is a 224-neuron fully connected layer, which receives three types of feature inputs: time series difference of the attacked measurement value, innovation difference and predicted measurement time series change.

3. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 2, wherein: The time series difference of the attacked measurement value is The innovation gap is The predicted measurement time series change is Among them, z k It represents the measurement value affected by the error data injection attack at the kth moment, and its expression is: Among them, y k is the true measurement value without attack, a k and m k are additive and multiplicative attack data respectively, and is a Bernoulli random variable, representing whether the attack occurs, where 1 indicates it occurs and 0 indicates it does not occur. is the predicted measurement value of EKF, which is predicted by the state Calculated by measuring the function h(·): The input layer maps the input parameters to a higher dimension by combining a fully connected layer with 224 neurons and a ReLU nonlinear activation function to perform feature space mapping; 58-unit unidirectional GRU layer to capture the dynamic temporal dependencies of the measurement data; The output layer is a fully connected layer with the same dimension as the state, and uses linear activation to output the residual correction Δy k (θ1), the observation residual correction network dynamically learns the attack interference pattern and outputs the corrected residual Δy k (θ1).

4. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 3, wherein: Two independent neural networks are used to replace the calculation of observation residuals and observation residual covariance matrices in the traditional extended Kalman filter, respectively, to construct a dual neural network correction architecture, in which: Design another GRU network output residual covariance matrix inverse The specific network structure has four layers, namely, fully connected input layer, GRU time series modeling layer, fully connected output layer and positive definite matrix output layer.

5. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 4, wherein: Design another GRU network output residual covariance matrix inverse The specific network structure has four layers, namely, fully connected input layer, GRU time series modeling layer, fully connected output layer and positive definite matrix output layer. The input layer is a 560-neuron fully connected layer that receives Δz k and the prior state covariance matrix P of EKF k|k-1 Three types of feature inputs are mapped into feature space through ReLU nonlinear activation function to map the input parameters to a higher dimension; 58-unit unidirectional GRU layer to capture the dynamic temporal dependencies of the measurement data; The output layer is a fully connected layer with the same dimension as the state, and the non-positive definite matrix A is generated; The positive definite matrix output layer is passed Where η is a small positive scalar, I m Represents the identity matrix of dimension m, where m is the observation dimension.

6. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 5, characterized in that: The dual neural network correction architecture is embedded in the EKF framework. The neural network outputs the corrected residual and the inverse of the covariance matrix to update the posterior state estimate and integrate the hybrid filter model. The specific steps include: The prediction step of the EKF is retained, and only the update step is enhanced by the neural network. The posterior state estimate is calculated as follows: Among them, P k|k (θ2) represents the posterior state covariance matrix, and n is the state dimension.

7. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 6, wherein: Two loss functions are designed based on the available dataset types, and an alternating training strategy is used to optimize the neural network parameters. The specific steps include: When the latent state data is available, the mean square error loss function loss1 is used: Among them, D1 consists of L trajectories of the true state and the attacked measurement value with a length of T. is represented as the true state value of the kth moment of the lth trajectory, and is the resulting posterior state.

8. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 7, wherein: Two loss functions are designed based on the available dataset types, and an alternating training strategy is used to optimize the neural network parameters. The specific steps include: When the original observation data is available, the measurement residual loss function loss2 is used: Among them, D2 consists of L trajectories of unattacked and attacked measurement values ​​of length T, is After the state transition function f(·) and the observation function, h(·) is obtained:

9. The neural network-enhanced Kalman filtering method for resisting error data injection according to claim 8, wherein: Two loss functions are designed based on the available dataset types, and an alternating training strategy is used to optimize the neural network parameters. The specific steps include: When updating θ1, the gradient is calculated and freeze θ2; When updating θ2, the gradient is calculated And freeze θ1.

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