A distributed filtering method, system, and storage medium based on depth expansion

Abstract

This invention discloses a distributed filtering method, system, and storage medium based on deep unfolding, relating to the field of sensor network technology. It aims to address the problems of poor adaptability to various noise levels, insufficient node estimation consistency, and high computational complexity in traditional methods. By establishing a network topology and system model, designing an information fusion filter structure, mapping the system iterative equations to a deep unfolded neural network, constructing a composite loss function containing estimation error and node divergence, and using the deep neural network for training to solve for the optimal gain, this invention is adaptable to noisy scenarios, achieving both high accuracy and consistency, with low computational complexity and strong robustness, making it suitable for fields such as environmental monitoring and dynamic target tracking.

Description

Technical Field

[0001] This invention relates to the field of sensor network technology, and in particular to a distributed filtering method, system, and storage medium based on depth unfolding. Background Technology

[0002] Sensor networks, leveraging their distributed sensing advantages, have enabled real-time monitoring of the state of complex systems across multiple domains. Distributed consensus filtering, as a core technology, requires information exchange between nodes to achieve accurate and consistent state estimation, thus resisting single points of failure and communication interference. However, existing technologies still suffer from the following significant drawbacks:

[0003] First, traditional filtering methods (such as H∞ filtering and variance-constrained filtering) are mostly designed for single types of noise, making it difficult to adapt to the complex environment where Gaussian noise and energy-bounded noise coexist in real-world scenarios. Furthermore, they do not fully integrate the measurement signals of neighboring nodes, relying solely on estimated values ​​for information exchange, which limits the estimation accuracy. For example, H∞ filtering is only applicable to energy-bounded noise, and variance-constrained filtering requires knowledge of the Gaussian noise covariance matrix, resulting in insufficient versatility.

[0004] Secondly, the problem of co-optimizing node consistency and estimation accuracy has not been effectively solved. Existing methods often prioritize minimizing estimation error, ignoring estimation discrepancies between nodes, making it difficult for the system to form a unified decision. Especially when the communication topology is complex, the accumulation of discrepancies will seriously reduce the robustness of the system.

[0005] Third, solving for the filter gain faces high computational complexity. Traditional methods require calculating the gain through linear matrix inequalities or recursive covariance, which involves a large number of matrix operations and is limited by the solvability of non-convex optimization problems, making it difficult to meet real-time requirements.

[0006] Although deep learning technology has been applied to the field of control, there is still a lack of effective solutions for combining it with distributed consensus filtering, particularly in constructing a network structure that adapts to the iterative equations and designing training objectives that balance accuracy and consistency. Therefore, there is an urgent need for a distributed filtering method that combines versatility, high accuracy, and low complexity. Summary of the Invention

[0007] To address the above technical problems, this invention provides a distributed filtering method based on depth expansion, comprising the following steps:

[0008] Step S1: Establish mathematical models of sensor networks and systems, and perform topological modeling of sensor networks and discrete-time linear systems respectively;

[0009] Step S2: Design the information fusion filter structure. Based on the information received by node i from neighbor node j, construct an intermediate estimator; and perform distributed information fusion. Node i fuses the intermediate estimates of all neighbor nodes and performs local filter updates.

[0010] Step S3: Construct a deep unfolded neural network model. First, unfold the system state equation and filter update equation into a multi-layer feedforward neural network by time step. Each layer corresponds to a time step k, and the neurons in the layer correspond to the system state and the estimated values ​​of each node. The network input is disturbance and noise, and the training parameter is the filter gain.

[0011] Step S4: Generate training data and perform incremental training. After training converges, output the optimal filter gain and substitute it into the filter update equation to achieve distributed consensus filtering.

[0012] The technical solution further defined in this invention is:

[0013] Furthermore, in step S1, sensor network topology modeling is performed using a directed graph. Describe the communication relationships between nodes, where Let N represent the set of nodes, and let N represent the total number of sensor nodes in the sensor network. Denotes the set of edges. Represents the adjacency matrix. This indicates that node i can receive information from its neighbor node j, and satisfies the following conditions: The neighbor set of node i is .

[0014] As described above, in a distributed filtering method based on depth expansion, step S1 involves modeling a discrete-time linear system. The system state equation and the sensor measurement equation are as follows:

[0015] ;

[0016] in, Indicates the system status. This represents the measurement signal of sensor node i. Indicates external disturbance. The noise level is represented by k, which represents the discrete time step. and Represents the system parameter matrix. and The parameter matrix representing sensor node i; , , as well as Representing states respectively Measurement signals External disturbances and measuring noise Dimensions.

[0017] As described above in a distributed filtering method based on depth expansion, in step S2, an intermediate estimator is constructed:

[0018] ;

[0019] in, This represents the state estimate of node j. This represents the measurement signal at node j. The parameter matrix representing sensor node j, Let represent the filter gain to be determined. If node i and node j have no communication connection, then... .

[0020] As described above in the distributed filtering method based on depth expansion, in step S2, the local filter update equation is:

[0021] ;

[0022] in, This represents the estimated state of node i at the next time step.

[0023] As described above in a distributed filtering method based on depth expansion, in step S3, the loss function is constructed:

[0024] ;

[0025] in, This represents the estimation error vector. This represents the estimation error vector of N nodes. Stacked vertically in sequence, forming a dimension of The overall column vector e(k), and ; Represents the node branching function. Describe an N-order diagonal matrix. and Let i be the in-degree and out-degree of node i, respectively. This represents the weighting coefficient used to balance the accuracy and consistency of the estimation.

[0026] As described above, in a distributed filtering method based on depth expansion, step S4 generates M sets of training data based on the statistical characteristics of perturbation and noise. Statistical properties include the covariance of Gaussian noise and the upper bound of energy-bounded noise.

[0027] As described above, in a distributed filtering method based on depth expansion, step S4 employs an incremental learning strategy, training the network layer by layer starting from k=1; firstly, the initial system state x(0) and the initial filter state are initialized. Then, forward propagation is used to calculate the system state x(k) and its estimate at each time step. Next, the loss function is calculated, and the filter gain is updated via backpropagation using the Adam optimizer. Finally, repeat the above steps until the K-layer network is trained.

[0028] A distributed filtering system based on depth expansion, comprising:

[0029] The model building module is used to build sensor network topology models, discrete-time linear system models, and measurement models, and to store adjacency matrices. With system parameter matrix ;

[0030] The filter design module includes an intermediate estimator and an information fusion unit, which are used to perform intermediate estimation calculation and neighbor information fusion, respectively, and output local state estimates.

[0031] The deep unfolded network module is used to map the system and filter iterative equations into a multi-layer neural network, define the loss function, and store training data and network parameters.

[0032] The training optimization module is used to train the network using incremental learning and stochastic gradient descent algorithms and solve for the optimal filter gain.

[0033] The filtering execution module is used to load the optimal filtering gain, update the state estimates of each node in real time, and output the estimation results.

[0034] A storage medium storing a computer program, which, when read and executed, implements the aforementioned distributed filtering method based on depth expansion.

[0035] The beneficial effects of this invention are:

[0036] (1) In this invention, it is not limited to a single noise type. Only the basic statistical characteristics of the noise (such as the covariance of Gaussian noise, the upper bound of energy of energy-bounded noise, the mean of exponential noise, etc.) need to be obtained to generate a training dataset that conforms to the actual interference law. Through the powerful learning ability of deep unfolded neural networks, the filter can automatically adapt to the distribution characteristics and interference intensity of different noises. Whether it is a single noise scene or a scene with multiple noise superposition, it can maintain stable filtering performance. This high versatility means that this invention does not need to redesign or adjust the core algorithm for specific noises. It can be directly applied to different noise environments such as environmental monitoring, industrial control, and target tracking, which greatly reduces the adaptation difficulty and maintenance cost in engineering applications. The scope of application covers the state estimation needs of most discrete-time linear systems.

[0037] (2) In this invention, the synergistic optimization of accuracy and consistency is achieved through two core designs: On the one hand, an innovative information fusion filter structure is designed, which not only fuses the state estimates of neighboring nodes, but also directly introduces the measurement signals of neighboring nodes. The measurement signals, as direct observations of the system state, contain original information that is closer to the real state. Compared with the traditional structure that only relies on the estimates, this effectively reduces the propagation and accumulation of estimation errors. On the other hand, a composite loss function containing estimation errors and node divergences is constructed, and the weighting coefficients are used to achieve the desired result. By flexibly balancing the priorities of both, the deviation between the estimated values ​​of each node and the true state and the degree of inconsistency between the estimated values ​​of the nodes are simultaneously optimized during the training process. Experimental verification shows that, under the same noise environment, the mean square error of the present invention is lower than that of traditional H∞ filtering and variance constraint filtering, and the average inconsistency is reduced. Especially in the dynamic target tracking scenario, the estimated values ​​of each node for the target position, velocity and acceleration can quickly converge to a consistent value, and the deviation from the true state is stable below 0.2. This not only ensures the estimation accuracy of individual nodes, but also achieves consensus among all nodes in the network, providing a reliable and unified basis for the global decision-making of the system.

[0038] (3) In this invention, the gain solution logic is completely reconstructed by deep unfolding technology. The system state equation and the filter iteration equation are mapped to a multi-layer feedforward neural network by time step. The original complex non-convex filter gain solution problem is transformed into a neural network parameter training problem. During the training process, relying on mature deep learning frameworks such as PyTorch, the Adam optimizer is used to implement stochastic gradient descent. The network parameters are trained layer by layer through incremental learning strategy. There is no need to manually solve complex linear matrix inequalities or recursive covariance equations, which greatly reduces the complexity of the calculation process and the cost of manual intervention. At the same time, the parallel computing characteristics of the neural network enable the training process to make efficient use of hardware resources. After the training is completed, the actual filtering stage only needs to call the pre-trained optimal gain. The state estimation update can be completed through simple matrix operations. The response speed meets the millisecond-level real-time requirements, which perfectly solves the pain points of traditional methods in solving complex problems and poor real-time performance. It is especially suitable for distributed scenarios with limited sensor node computing resources.

[0039] (4) In this invention, the robustness and fault tolerance of the system are significantly enhanced through the dual design of distributed structure and redundant information fusion: the distributed structure enables each node to have independent filtering and information processing capabilities, without the dependence on a core node. Even if some nodes fail and stop working, the remaining nodes can still complete the state estimation through their own measurements and information interaction with other neighboring nodes, without causing the global system to be paralyzed; the redundant information fusion mechanism ensures that each node can receive measurement signals and estimated values ​​from multiple neighboring nodes, forming a multi-source information backup. Even if a communication link is interrupted or the estimated value of a single neighboring node is distorted by interference, the node can still complete the state estimation through its own measurements and information interaction with other neighboring nodes, without causing the global system to be paralyzed; By integrating signal correction errors from other reliable information sources, the impact of a single information source failure on filtering performance is avoided. Experimental verification shows that in scenarios with 1-2 node failures or communication link interruptions, the mean square estimation error fluctuation of this invention does not exceed 0.05, and the average inconsistency fluctuation does not exceed 0.03, maintaining stable estimation accuracy and node consistency. In contrast, traditional methods often experience a 2-3 fold increase in mean square estimation error under similar failure scenarios, and may even exhibit filter divergence. This strong robustness enables this invention to adapt to complex and ever-changing engineering environments, effectively resisting sudden problems such as node failures and communication interference, and ensuring long-term stable operation of the system. Attached Figure Description

[0040] Figure 1 This is a schematic diagram of the overall process of the present invention;

[0041] Figure 2 This is a schematic diagram of the information fusion filter in an embodiment of the present invention;

[0042] Figure 3 This is a comparison chart of the mean square error between the method of this invention and H∞ filtering;

[0043] Figure 4 This is a comparison chart of the average inconsistency between the method of this invention and H∞ filtering;

[0044] Figure 5 This is a diagram showing the position estimation results of the method of the present invention in dynamic target tracking. Detailed Implementation

[0045] This embodiment provides a distributed filtering method based on depth expansion, such as... Figure 1 As shown, it includes the following steps:

[0046] Step S1: Establish mathematical models of sensor networks and systems, and perform topological modeling of sensor networks and discrete-time linear systems respectively.

[0047] Sensor network topology modeling: using directed graphs Describe the communication relationships between nodes, where Let N represent the set of nodes, and let N represent the total number of sensor nodes in the sensor network. Denotes the set of edges. Represents the adjacency matrix. This indicates that node i can receive information from its neighbor node j, and satisfies the following conditions: The neighbor set of node i is .

[0048] Discrete-time linear system modeling: The system state equation and sensor measurement equation are as follows:

[0049] ;

[0050] in, Indicates the system status. This represents the measurement signal of sensor node i. Indicates external disturbance. The noise level is represented by k, which represents the discrete time step. and Represents the system parameter matrix. and The parameter matrix representing sensor node i; , , as well as Representing states respectively Measurement signals External disturbances and measuring noise Dimensions.

[0051] Step S2, Design as follows Figure 2The information fusion filter structure shown constructs an intermediate estimator based on the information received by node i from its neighbor node j; and performs distributed information fusion, where node i fuses the intermediate estimates from all its neighbors to update the local filter.

[0052] Intermediate Estimator Design: For the information received by node i from its neighbor node j, construct an intermediate estimator:

[0053] ;

[0054] in, This represents the state estimate of node j. This represents the measurement signal at node j. The parameter matrix representing sensor node j, Let represent the filter gain to be determined. If node i and node j have no communication connection, then... .

[0055] Distributed information fusion: Node i fuses the intermediate estimates from all its neighboring nodes to obtain the local filter update equation:

[0056] ;

[0057] in, This represents the estimated state of node i at the next time step.

[0058] Step S3: Construct a deep unfolded neural network model.

[0059] First, network structure mapping is performed: the system state equation and filter update equation are expanded into a multi-layer feedforward neural network over time steps, with each layer corresponding to one time step k, and neurons within the layer corresponding to the system state and the estimated values ​​of each node; the network input is the perturbation. With noise The training parameter is the filter gain. .

[0060] Loss function design: Balancing estimation accuracy and node consistency, the loss function is constructed as follows:

[0061] ;

[0062] in, This represents the estimation error vector. This represents the estimation error vector of N nodes. Stacked vertically in sequence, forming a dimension of The overall column vector e(k), and ; Represents the node branching function. Describe an N-order diagonal matrix. and Let i be the in-degree and out-degree of node i, respectively. This represents the weighting coefficient, used to balance the accuracy and consistency of the estimation.

[0063] Step S4: Generate training data and perform incremental training. After training converges, output the optimal filter gain and substitute it into the filter update equation to achieve distributed consensus filtering.

[0064] Training data generation: Based on the statistical properties of perturbation and noise (such as the covariance of Gaussian noise and the upper bound of energy-bounded noise), generate M sets of training data. .

[0065] Incremental training: An incremental learning strategy is adopted, and the network is trained layer by layer starting from k=1; firstly, the initial state of the system x(0) and the initial state of the filter are initialized. Then, forward propagation is used to calculate the system state x(k) and its estimate at each time step. Next, the loss function is calculated, and the filter gain is updated via backpropagation using the Adam optimizer. Finally, repeat the above steps until the K-layer network is trained.

[0066] In this embodiment, the sensor network has N=5 nodes, and the adjacency matrix is ​​as follows:

[0067] .

[0068] Configure system parameters: , , , Randomly generated stable Next, adjust the noise settings: It follows an exponential distribution (mean 1). The network follows a standard normal distribution; then the network parameters are set as follows: number of training steps K=50, number of training datasets M=30, and learning rate 0.01.

[0069] In practice, the model building module is first initialized by importing the network topology and system parameters; the filter design module constructs the intermediate estimator and information fusion unit according to the above technical solution, and initializes the filter gain. The values ​​are random small values; then the deep expansion network module generates training data, the loss function is defined, and a neural network with K=50 layers is constructed.

[0070] Then the training optimization module performs incremental training: the first round of training trains k=1 layer, generating... Calculate x(1) forward. Reverse update Then, the number of layers is increased sequentially until K=50, and the training results of the previous layer are reused as initial parameters in each round; after training converges, the filtering execution module loads the optimal gain. It receives sensor measurement signals in real time and outputs state estimates for each node.

[0071] Finally, experimental verification was conducted: such as... Figure 3 and Figure 4 As shown, compared with the H∞ filter, when the upper bound of the noise energy is 100, the mean square error of the filter in this embodiment is reduced by 58% and the inconsistency is reduced by 62% compared with the H∞ filter, demonstrating a significant performance advantage under strong noise. Figure 5 The figure shown is the position estimation result of the method in this embodiment in dynamic target tracking. For target state estimation containing position, velocity and acceleration, the mean square error is stable below 0.2 and the inconsistency is less than 0.1, and the tracking accuracy meets the actual requirements.

[0072] This embodiment also provides a distributed filtering system based on depth expansion, including:

[0073] The model building module is used to build sensor network topology models, discrete-time linear system models, and measurement models, and to store adjacency matrices. With system parameter matrix ;

[0074] The filter design module includes an intermediate estimator and an information fusion unit, which are used to perform intermediate estimation calculation and neighbor information fusion, respectively, and output local state estimates.

[0075] The deep unfolded network module is used to map the system and filter iterative equations into a multi-layer neural network, define the loss function, and store training data and network parameters.

[0076] The training optimization module is used to train the network using incremental learning and stochastic gradient descent algorithms and solve for the optimal filter gain.

[0077] The filtering execution module is used to load the optimal filtering gain, update the state estimates of each node in real time, and output the estimation results.

[0078] The method in this embodiment has strong adaptability to various noise types: Traditional distributed filtering methods have significant limitations in terms of noise types. For example, H∞ filtering can only adapt to energy-bounded noise, and variance-constrained filtering requires precise knowledge of the covariance matrix of Gaussian noise in advance. When faced with scenarios in industrial environments where multiple types of interference coexist, such as Gaussian noise, energy-bounded noise, exponential noise, and uniform noise, it is often necessary to change different algorithms or redesign the filtering structure, resulting in high adaptation costs and extremely poor versatility.

[0079] This embodiment of the method completely overcomes this limitation. Its core advantage lies in the fact that it is not limited to a single noise type. It only requires obtaining the basic statistical characteristics of the noise (such as the covariance of Gaussian noise, the upper bound of energy for energy-bounded noise, and the mean of exponential noise) to generate a training dataset that conforms to the actual interference patterns. Through the powerful learning capabilities of deep neural networks, the filter can automatically adapt to the distribution characteristics and interference intensity of different noises, maintaining stable filtering performance in both single-noise scenarios and scenarios with multiple noise superpositions. This high versatility means that this embodiment of the method does not require redesigning or adjusting the core algorithm for specific noises. It can be directly applied to scenarios with different noise environments, such as environmental monitoring, industrial control, and target tracking, significantly reducing the adaptation difficulty and maintenance costs in engineering applications. Its applicability covers the state estimation needs of most discrete-time linear systems.

[0080] The method in this embodiment has both high estimation accuracy and consistency: Existing distributed filtering methods generally suffer from the design flaw of emphasizing accuracy while neglecting consensus. Most algorithms only aim to minimize the estimation error and ignore the consistency of the estimation values ​​of each node in the sensor network. This leads to the accumulation of estimation discrepancies between nodes even if some nodes have high estimation accuracy, which in turn affects the reliability of the system's global decision-making. Especially when the communication topology is complex or the signals of some nodes are interfered with, the discrepancies may prevent the system from forming a unified and effective conclusion.

[0081] This embodiment achieves synergistic optimization of accuracy and consistency through two core design features: First, it innovatively designs an information fusion filter structure that not only fuses the state estimates of neighboring nodes but also directly incorporates the measurement signals of neighboring nodes. These measurement signals, as direct observations of the system state, contain raw information closer to the true state, effectively reducing the propagation and accumulation of estimation errors compared to traditional structures that rely solely on estimates. Second, it constructs a composite loss function that includes both estimation errors and node divergences, using weighting coefficients... By flexibly balancing the priorities of both, the deviation between the estimated values ​​of each node and the true state, as well as the degree of inconsistency between the estimated values ​​of nodes, are simultaneously optimized during training. Experimental verification shows that, under the same noise environment, the mean squared error of the method in this embodiment is lower than that of traditional H∞ filtering and variance constraint filtering, and the average inconsistency is reduced. Especially in dynamic target tracking scenarios, the estimated values ​​of target position, velocity, and acceleration of each node can quickly converge to a consensus, and the deviation from the true state is stable below 0.2. This ensures the estimation accuracy of individual nodes and achieves consensus among all nodes in the network, providing a reliable and unified basis for global system decision-making.

[0082] The method in this embodiment has low computational complexity: The core bottleneck of traditional distributed filtering methods lies in the high computational complexity of the filter gain calculation process. For example, H∞ filtering needs to be transformed into an optimization problem of linear matrix inequalities, while variance-constrained filtering relies on recursive covariance calculation. These methods not only involve a large number of high-dimensional matrix multiplications and inversions, but are also limited by the convergence of non-convex optimization problems. They often consume a lot of computational resources and are difficult to meet the requirements of high response speed in scenarios such as real-time monitoring of sensor networks and dynamic target tracking.

[0083] This embodiment completely reconstructs the gain calculation logic through deep decomposition technology, mapping the system state equation and filter iteration equation to a multi-layer feedforward neural network over time steps. The originally complex non-convex filter gain calculation problem is transformed into a neural network parameter training problem. During training, relying on mature deep learning frameworks such as PyTorch, the Adam optimizer is used to implement stochastic gradient descent, and the network parameters are trained layer by layer through an incremental learning strategy. There is no need to manually solve complex linear matrix inequalities or recursive covariance equations, which significantly reduces the complexity of the computation process and the cost of manual intervention. At the same time, the parallel computing characteristics of the neural network enable efficient use of hardware resources during training. After training, the actual filtering stage only needs to call the pre-trained optimal gain, and the state estimation update can be completed through simple matrix operations. The response speed meets the millisecond-level real-time requirements, perfectly solving the pain points of traditional methods such as complexity and poor real-time performance. It is especially suitable for distributed scenarios with limited sensor node computing resources.

[0084] The method in this embodiment has high robustness: Traditional distributed filtering systems have weak robustness. On the one hand, the centralized or semi-distributed structure makes the system sensitive to single-point failures. Once a core node or critical communication link is interrupted, the local or even global filtering performance will plummet. On the other hand, most methods rely on only a single information source (such as their own measurement or the estimate of a single neighbor), resulting in low information redundancy. When faced with sudden situations such as node failure, communication interruption, and signal interference, they lack effective fault tolerance and anti-interference mechanisms.

[0085] This embodiment significantly enhances the robustness and fault tolerance of the system through a dual design of distributed structure and redundant information fusion. The distributed structure enables each node to have independent filtering and information processing capabilities, without dependence on a core node. Even if some nodes fail and stop working, the remaining nodes can still complete state estimation through their own measurements and information interaction with other neighboring nodes, preventing the global system from paralyzing. The redundant information fusion mechanism ensures that each node can receive measurement signals and estimates from multiple neighboring nodes, forming multi-source information backup. Even if a communication link is interrupted or the estimate of a single neighboring node is distorted by interference, the node can still correct the error by fusing signals from other reliable information sources, avoiding the impact of a single information source failure on filtering performance. Experimental verification shows that in scenarios with 1-2 node failures or communication link interruptions, the mean square estimation error of this embodiment does not fluctuate by more than 0.05, and the average inconsistency fluctuation does not exceed 0.03, maintaining stable estimation accuracy and node consistency. In contrast, traditional methods often experience a 2-3 fold increase in mean square estimation error under similar failure scenarios, and may even exhibit filter divergence. This strong robustness enables the method in this embodiment to adapt to complex and ever-changing engineering environments, effectively resist sudden problems such as node failures and communication interference, and ensure the long-term stable operation of the system.

[0086] In addition to the embodiments described above, the present invention may have other implementations. All technical solutions formed by equivalent substitution or equivalent transformation fall within the protection scope claimed by the present invention.

Claims

1. A distributed filtering method based on depth expansion, characterized in that: Includes the following steps: Step S1: Establish mathematical models of sensor networks and systems, and perform topological modeling of sensor networks and discrete-time linear systems respectively; Step S2: Design the information fusion filter structure. Based on the information received by node i from neighbor node j, construct an intermediate estimator; and perform distributed information fusion. Node i fuses the intermediate estimates of all neighbor nodes and performs local filter updates. Step S3: Construct a deep unfolded neural network model. First, unfold the system state equation and filter update equation into a multi-layer feedforward neural network by time step. Each layer corresponds to a time step k, and the neurons in the layer correspond to the system state and the estimated values ​​of each node. The network input is disturbance and noise, and the training parameter is the filter gain. Step S4: Generate training data and perform incremental training. After training converges, output the optimal filter gain and substitute it into the filter update equation to realize distributed consensus filtering. In step S2, an intermediate estimator is constructed: ； in, This represents the state estimate of node j. This represents the measurement signal at node j. The parameter matrix representing sensor node j, Let represent the filter gain to be determined. If node i and node j have no communication connection, then... ; In step S2, the local filter update equation is: ； in, This represents the estimated state of node i at the next time step. In step S3, the loss function is constructed: ； in, This represents the estimation error vector. This represents the estimation error vector of N nodes. Stacked vertically in sequence, forming a dimension of The overall column vector e(k), and ; Represents the node branching function. Describe an N-order diagonal matrix. and Let i be the in-degree and out-degree of node i, respectively. This represents the weighting coefficient used to balance the accuracy and consistency of the estimation.

2. The distributed filtering method based on depth expansion according to claim 1, characterized in that: In step S1, sensor network topology modeling is performed using a directed graph. Describe the communication relationships between nodes, where Let N represent the set of nodes, and let N represent the total number of sensor nodes in the sensor network. Denotes the set of edges. Represents the adjacency matrix. This indicates that node i can receive information from its neighbor node j, and satisfies the following conditions: The neighbor set of node i is .

3. The distributed filtering method based on depth expansion according to claim 2, characterized in that: In step S1, a discrete-time linear system model is performed, and the system state equation and sensor measurement equation are as follows: ； in, Indicates the system status. This represents the measurement signal of sensor node i. Indicates external disturbance. The noise level is represented by k, which represents the discrete time step. and Represents the system parameter matrix. and The parameter matrix representing sensor node i; , , as well as Representing states respectively Measurement signals External disturbances and measuring noise Dimensions.

4. The distributed filtering method based on depth expansion according to claim 3, characterized in that: In step S4, M sets of training data are generated based on the statistical characteristics of the perturbation and noise. ; Statistical properties include the covariance of Gaussian noise and the upper bound of energy-bounded noise.

5. The distributed filtering method based on depth expansion according to claim 4, characterized in that: In step S4, an incremental learning strategy is adopted to train the network layer by layer starting from k=1; firstly, the initial state of the system x(0) and the initial state of the filter are initialized. Then, forward propagation is used to calculate the system state x(k) and its estimate at each time step. Next, the loss function is calculated, and the filter gain is updated via backpropagation using the Adam optimizer. Finally, repeat the above steps until the K-layer network is trained.

6. A distributed filtering system based on depth expansion, used to perform the method according to any one of claims 1-5, characterized in that: include: The model building module is used to build sensor network topology models, discrete-time linear system models, and measurement models, and to store adjacency matrices. With system parameter matrix ; The filter design module includes an intermediate estimator and an information fusion unit, which are used to perform intermediate estimation calculation and neighbor information fusion, respectively, and output local state estimates. The deep unfolded network module is used to map the system and filter iterative equations into a multi-layer neural network, define the loss function, and store training data and network parameters. The training optimization module is used to train the network using incremental learning and stochastic gradient descent algorithms and solve for the optimal filter gain. The filtering execution module is used to load the optimal filtering gain, update the state estimates of each node in real time, and output the estimation results.

7. A storage medium, characterized in that: The storage medium stores a computer program, which, when read and executed, implements the distributed filtering method based on depth expansion as described in any one of claims 1-5.

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