Adaptive state estimation method based on KF and PINN deep fusion

Abstract

The invention belongs to the field of navigation, and discloses a KF and PINN deep fusion-based adaptive state estimation method, which comprises the following steps of: establishing a discrete time state equation and a measurement equation of Kalman filtering, and setting initial conditions of a filter; constructing a neural network model based on KF and PINN; training a neural network model; utilizing the trained neural network model to predict statistical characteristics of noise to obtain a noise covariance matrix at the moment k; constructing a filtering time updating process; constructing a filtering measurement updating process; and repeating the steps to obtain all state posteriori estimations. According to the method, the problem of estimation errors caused by inaccurate noise covariance of traditional Kalman filtering is effectively solved, and the precision and robustness of state estimation are remarkably improved.

Description

Technical Field

[0001] The present invention relates to the field of navigation, and in particular, to an adaptive state estimation method based on the deep fusion of KF and PINN. Background Art

[0002] In the field of state estimation, the Kalman Filter (KF) has been widely used in unmanned driving and robot navigation systems due to its high efficiency and optimality. However, the performance of traditional Kalman filtering depends on accurate system models and noise statistical characteristics. In practical applications, the system dynamics are often highly nonlinear and even contain outliers, resulting in a significant decline in the performance of traditional Kalman filtering. To enhance the robustness of state estimation, adaptive Kalman filtering methods have evolved as an effective solution, which dynamically adjusts the noise covariance matrix to respond to changes in system dynamics and noise intensity. Threshold-based adaptive filtering is a commonly used method, which has the advantages of simple implementation and small computational complexity. However, the selection of the threshold requires experience and it is difficult to adapt to situations where the noise changes violently. In addition, traditional adaptive filtering methods still have obvious deficiencies in dealing with nonlinearity and outliers. With the rapid development of deep learning, neural network algorithms have shown great potential in processing multi-dimensional time series. Physics-Informed Neural Networks (PINN) can effectively model the dynamic characteristics of complex systems by embedding physical laws into the loss function of neural networks. Neural networks can learn the internal relationship between system states and measurement data, thereby extracting relevant spatio-temporal features and predicting the dynamic changes of interference. Therefore, the present invention proposes an adaptive state estimation method based on the deep fusion of KF and PINN. First, based on the PINN network, the system dynamics and noise characteristics are modeled, and the key parameters of the Kalman filter (such as the measurement noise covariance matrix) are output to adjust the noise covariance matrix of the Kalman filter in real time, so as to achieve high-precision state estimation in a complex noise environment. Secondly, the optimal gain condition of the Kalman filter is used as a physical constraint to be embedded in the loss function of the neural network, realizing the joint optimization of physical laws and data-driven. Finally, an adaptive weight strategy is used to dynamically balance the data fitting error and the physical constraint residual, achieving high-precision and robust state estimation. This method can effectively handle nonlinear and outlier interferences, and at the same time has real-time and self-adaptability, and is applicable to the state estimation tasks of complex dynamic systems. Summary of the Invention

[0003] The purpose of the present invention is to provide an adaptive state estimation method based on the deep fusion of KF and PINN to solve the problems raised in the above background art.

[0004] To achieve the above purpose, the present invention provides the following technical solutions:

[0005] An adaptive state estimation method based on the deep fusion of KF and PINN, comprising:

[0006] Step 1, establish the discrete-time state equation and measurement equation of the Kalman filter, and set the initial conditions of the filter;

[0007] Step 2, construct a neural network model based on KF and PINN. The neural network model consists of a GRU network layer, a linear layer, and a Kalman filter layer. By learning the dynamic characteristics of the measurement noise, the measurement noise covariance matrix required for the Kalman filter is output The Kalman filter layer is based on the dynamically adjusted Perform state prediction and update, and finally output the filtered posterior estimation state

[0008] Step 3, train the neural network model: The predicted by the GRU network layer is passed to the Kalman filter layer, and the state update process of the filter is completed by dynamically adjusting to realize the forward propagation of the network; Use the physics-informed neural network to embed the optimal gain condition of the Kalman filter as a physical constraint into the loss function, and construct a composite loss function L = L data + λL physics , where L data is the data fitting error, L physics is the residual norm of the optimal gain condition, and λ is the weight parameter; Calculate the gradient through backpropagation and optimize the network parameters to accurately determine the network parameters;

[0009] Step 4, use the trained neural network model to predict the statistical characteristics of the noise, and obtain the noise covariance matrix at time k

[0010] Step 5, construct the filtering time update process: Based on Step 1, obtain the predicted value of the state variable at time k and the predicted covariance P k|k-1 , and obtain the gain K k at time k using the optimal gain theorem;

[0011] Step 6, construct the filtering measurement update process: Based on Step 5, use the obtained measurement information y k and the optimal gain K k to complete the posterior estimation of the state, and calculate the covariance matrix P k|k ;

[0012] Step 7, repeatedly execute Step 4 - Step 6 to obtain all the posterior state estimations.

[0013] Furthermore, the models of the discrete-time state equation and measurement equation in Step 1 are as follows:

[0014]

[0015] Among them, is the system state vector (acceleration) at time k, F is a diagonalizable state matrix, ω k-1 obeys a Gaussian distribution with zero mean and covariance Q k-1 , H is the measurement matrix, y k is the system measurement value (acceleration measured by the system sensor), ε 1,k obeys a Gaussian distribution with zero mean and covariance R k , β k is a random variable following a 0-1 distribution, ε 2,k is an abnormal measurement, obeying a Gaussian distribution with zero mean and covariance .

[0016] Furthermore, in step 2, the GRU network layer takes the system measurement information as input, learns the dynamic characteristics of the measurement noise, and outputs the measurement noise covariance matrix required for Kalman filtering The linear layer connects the GRU network layer and the Kalman filtering layer, and transmits to the Kalman filtering layer.

[0017] Furthermore, in step 3, the optimal gain condition is:

[0018]

[0019] Among them, J(K k ) = tr(P k|k ), tr(P k|k ) represents taking the trace of the covariance matrix P k|k , H represents the measurement matrix, H T is the transpose matrix of H, and the loss function L physics is constructed according to formula (3), and its residual can be expressed as:

[0020]

[0021] The loss function L data is constructed, and the mean square error between the posterior estimated state and the true state can also be expressed as:

[0022]

[0023] Among them, x true is the label value.

[0024] Furthermore, in step 4, the measurement information y at time k kInput it into the neural network model trained in step 3 to predict and output the measurement noise covariance matrix at time k

[0025] Furthermore, in the said step 5, the predicted value at time k and the predicted covariance P k|k-1 can be given as:

[0026]

[0027] P k|k-1 = FP k-1|k-1 F T + Q k (8)

[0028] where Q k is the process noise covariance and F T is the transpose matrix of F

[0029] Furthermore, in the said step 5, the optimal gain K k|k is determined by minimizing the trace of the error covariance matrix P k , then:

[0030]

[0031] where argmin(·) is used to represent the independent variable value that makes the given function obtain the minimum value;

[0032] Derive J(K k ) = tr(P k|k ) with respect to K k , then:

[0033]

[0034] where J(·) represents the trace of the error covariance matrix of the state estimation;

[0035] Since P k|k-1 is a symmetric matrix, (HP k|k-1 ) T = P k|k-1 H T , therefore:

[0036]

[0037] Let obtain:

[0038]

[0039] After arrangement, the optimal Kalman gain is obtained:

[0040]

[0041] Further, in the said step 6, the measurement information is acquired by sensors in the state estimation system, and the measurement update process refers to obtaining the posterior state estimation and covariance, and the values of the posterior state estimation and covariance are calculated through the following formula:

[0042]

[0043] P k|k =(I-K k H)P k|k-1 (15)

[0044] where I is the identity matrix.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention proposes an adaptive state estimation method based on the deep fusion of KF and PINN. By combining the learning ability of the neural network with the optimal gain property of the Kalman filter, the advantages of the strong dynamic modeling ability of the neural network and the fast convergence speed of the Kalman filter are fully utilized. This method effectively solves the estimation error problem caused by inaccurate noise covariance in the traditional Kalman filter, and significantly improves the accuracy and robustness of state estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 : Flowchart of the adaptive state estimation method based on the deep fusion of KF and PINN;

[0047] Figure 2 : Schematic diagram of the GRU network structure;

[0048] Figure 3 : Block diagram of the state estimation with the deep fusion of KF and PINN. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0049] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0050] Referring to Figure 1 、 Figure 2 and Figure 3 , an adaptive state estimation method based on the deep fusion of KF and PINN includes the following steps:

[0051] Step 1, establish the discrete-time state equation and measurement equation of the Kalman filter:

[0052]

[0053] wherein, is the system state vector at time k, and F is a diagonalizable state matrix. ω k-1 obeys a Gaussian distribution with zero mean and covariance Q k-1 . H is the measurement matrix. y k is the system measurement value, and ε 1,k obeys a Gaussian distribution with zero mean and covariance R k . ε 2,k is the abnormal measurement and obeys a Gaussian distribution with zero mean and covariance .

[0054] Set the initial conditions of the filter: the initial state estimate x k = [0 0] T , and the initial covariance matrix P0 = I2 (I2 is a 2-dimensional identity matrix).

[0055] Step 2, construct a neural network model based on KF and PINN. The neural network model consists of a GRU network layer, a linear layer, and a Kalman filter layer.

[0056] The input of the GRU network layer is the acceleration measurement information y k (acceleration), and the time window length is 10. The output of the GRU network layer is the diagonal elements of the measurement noise covariance matrix . The input dimension of the network is 2, the hidden layer dimension is 64, and the output dimension is 2.

[0057] Linear layer: Connect the GRU network layer and the Kalman filter module, perform dimensional transformation and non-linear mapping on to ensure its parameter compatibility with the Kalman filter module.

[0058] The Kalman filter layer performs state prediction and update based on the dynamically adjusted , and outputs the filtered posterior estimate state and the predicted covariance P k|k .

[0059] Step 3, train the neural network model. Use the Xsens sensor to capture the high-precision acceleration information of the unmanned vehicle as the label value x true . Align the measurement data of the accelerometer with the label value to construct a training data set. According to the optimal gain theorem, we can obtain:

[0060]

[0061] wherein, J(K k ) = tr(P k|k ), tr(Pk|k ) represents taking the covariance matrix P k|k of the trace, H represents the measurement matrix, H T is the transpose matrix of H, and the loss function L is constructed according to formula (3) physics , and its residual can be expressed as:

[0062]

[0063] Construct the loss function L data , and the mean square error between the posterior estimated state and the true state can also be expressed as:

[0064]

[0065] Construct the composite loss function:

[0066]

[0067] where λ = 0.5 is the weight parameter, use the Adam optimizer, and the learning rate is 10 -3 , the batch size is 32, and the number of iterations is 200. Calculate the gradient through backpropagation and optimize the network parameters until the loss function converges.

[0068] Step 4, predict the statistical characteristics of the noise. Input the acceleration measurement information y at time k k into the trained KF-PINN network, and predict and output the measurement noise covariance matrix at time k

[0069] Step 5, construct the filtering time update process. Calculate the predicted value of the state variable at time k and the predicted covariance P k|k-1 :

[0070]

[0071] P k|k-1 = FP k-1|k-1 F T + Q k (8)

[0072] where Q k is the process noise covariance, and F T is the transpose matrix of F.

[0073] Determine the optimal gain K by minimizing the trace of the error covariance matrix P k|k , then: k

[0074]

[0075] ​Among them, argmin(·) is used to represent the independent variable value that makes the given function obtain the minimum value;

[0076] For J(K k ) = tr(P k|k ) with respect to K k take the derivative, then:

[0077]

[0078] Among them, J(·) represents the trace of the error covariance matrix of state estimation;

[0079] Since P k|k-1 is a symmetric matrix, (HP k|k-1 ) T = P k|k-1 H T , so:

[0080]

[0081] Let obtain:

[0082]

[0083] After arrangement, the optimal Kalman gain is obtained:

[0084]

[0085] Step 6, construct the filtering measurement update process. Use the acceleration information y k and the optimal gain K k to calculate the posterior state estimation at time k and the covariance matrix P k|k :

[0086]

[0087] P k|k = (I - K k H)P k|k-1 (15).

[0088] Step 7, repeat Step 4 to Step 6, and process the acceleration measurement information at each moment in turn to obtain the estimated acceleration sequence at all moments.

[0089] As Figure 2 shown, GRU is an improved recurrent neural network (RNN), which controls the flow of information by introducing an update gate z t and a reset gate r t and effectively solves the problems of gradient vanishing and gradient explosion in the training of traditional RNN for long sequences. z tDetermine how much information in the current state comes from the hidden state of the previous moment, r t Control how much information in the hidden state of the previous moment needs to be forgotten. Finally, by combining z t and r t 's output, generate the hidden state of the current moment and output the current predicted value h t . The GRU layer contains multiple hidden units and can effectively model and predict long-sequence data. The input at each time step is X t , and the final output is the hidden state h t , thus realizing the efficient processing and prediction of time-series data.

[0090] This embodiment details the specific implementation process of the adaptive state estimation method based on the deep fusion of KF and PINN in the acceleration estimation of an unmanned vehicle. By dynamically adjusting the measurement noise covariance matrix through the KF-PINN network and combining the optimal gain condition of the Kalman filter, high-precision and strong-robustness acceleration estimation is achieved. This method has broad application prospects in fields such as unmanned aerial vehicles and robot navigation.

[0091] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An adaptive state estimation method based on the deep fusion of KF and PINN, characterized in that, Including: Step 1: Establish the discrete-time state equation and measurement equation of the Kalman filter, and set the initial conditions of the filter; Step 2, construct a neural network model based on KF and PINN. The neural network model consists of a GRU network layer, a linear layer, and a Kalman filter layer. By learning the dynamic characteristics of the measurement noise, it outputs the measurement noise covariance matrix required for the Kalman filter. The Kalman filter layer is based on dynamically adjusted Perform state prediction and update, and finally output the filtered posterior estimation state. Step 3, training the neural network model: The prediction of the GRU network layer is passed to the Kalman filter layer, and the state update process of the filter is completed by dynamically adjusting to achieve the forward propagation of the network; Using the physics-informed neural network, the optimal gain condition of the Kalman filter is embedded in the loss function as a physical constraint to construct a composite loss function \(L = L data +\lambda L physics , where \(L data is the data fitting error, \(L physics is the residual norm of the optimal gain condition, and \(\lambda\) is the weight parameter; Calculate the gradient through backpropagation and optimize the network parameters to achieve the accurate determination of the network parameters; Step 4: Use the trained neural network model to predict the statistical characteristics of the noise and obtain the noise covariance matrix at time k Step 5, construct the filtering time update process: Based on Step 1, obtain the predicted value of the state variable at time k and the predicted covariance P k|k-1 , and use the optimal gain theorem to obtain the gain K at time k k ; Step 6, construct the filtering measurement update process: Based on Step 5, utilize the obtained measurement information y k and the optimal gain K k to complete the posterior estimation of the state and calculate the covariance matrix P k|k ; Step 7: Repeat Steps 4 - 6 to obtain all the posterior state estimates.

2. The adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, wherein, The models of the discrete-time state equation and measurement equation in Step 1 are as follows: Among them, is the system state vector at time k, F is the diagonalizable state matrix, ω k-1 obeys a Gaussian distribution with zero mean and covariance Q k-1 ; H is the measurement matrix, y k is the system measurement value, ε 1,k obeys a Gaussian distribution with zero mean and covariance R k ; β k is a random variable following a 0-1 distribution, ε 2,k is an abnormal measurement, obeying a Gaussian distribution with zero mean and covariance of .

3. An adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, characterized in that, In the said step 2, the GRU network layer takes the system measurement information as input, learns the dynamic characteristics of the measurement noise, and outputs the measurement noise covariance matrix required for the Kalman filter. The linear layer connects the GRU network layer and the Kalman filter layer and passes it to the Kalman filter layer.

4. An adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, characterized in that In Step 3, the optimal gain condition is: where, J(K k ) = tr(P k|k ), tr(P k|k ) represents taking the trace of the covariance matrix P k|k , H represents the measurement matrix, H T is the transpose matrix of H, and the loss function L physics is constructed according to formula (3), and its residual can be expressed as: Construct the loss function L data , and the mean square error between the posterior estimated state and the true state can also be expressed as: where x true is the tag value.

5. The adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, wherein In the said step 4, the measurement information y at time k k is input into the neural network model trained in step 3 to predict and output the measurement noise covariance matrix at time k 6. The adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, characterized in that In step 5, the predicted value at time k and the predicted covariance P k|k-1 can be given as: P k|k-1 = FP k-1|k-1 F T + Q k (8) Among them, Q k is the process noise covariance, and F T is the transpose matrix of F.

7. An adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, characterized in that, In the said step 5, the optimal gain K is determined by minimizing the trace of the error covariance matrix P k|k and then: k then: where argmin(·) is used to represent the independent variable value that makes the given function obtain the minimum value; Derivative of J(K k ) = tr(P k|k ) with respect to K k yields: where J(·) represents the trace of the error covariance matrix of the state estimate; Since P k|k-1 is a symmetric matrix, (HP k|k-1 ) T = P k|k-1 H T , thus: Let Obtain: After arrangement, the optimal Kalman gain is obtained:

8. An adaptive state estimation method based on the deep fusion of KF and PINN according to claim 1, characterized in that In Step 6, the measurement information is collected by the sensors in the state estimation system. The measurement update process refers to obtaining the posterior state estimate and covariance, and the values of the posterior state estimate and covariance are calculated by the following formula: P k|k = (I - K k H)P k|k-1 (15) where I is the identity matrix.

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