Recursive filtering method and system based on matrix decomposition in sparse sensor measurements

Abstract

The application provides a sensor measurement sparse case based on matrix decomposition recursive filtering method and system, belongs to the field of nonlinear complex network state recursive filtering. In order to solve the problems of sensor network data loss, high energy consumption, multiple noise changes and security in the existing complex network environment. The application includes a low duty cycle scheduling mechanism, which controls the activation time of the sensor node, ensures that the sensor node remains in a dormant state during the inactive period; Matrix decomposition technology is used to process sparse data and improve the processing capacity of incomplete or missing data; Dynamic parameter adjustment based on Adam optimization algorithm, adapt to different network noise environment and nonlinear data characteristics; DoS defense mechanism, using a random variable model to identify abnormal fluctuations in the data, real-time monitoring and isolation of potential attack data; Recursive error covariance matrix update, realizing data gradual update and accurate filtering.

Description

Technical Field

[0001] This invention relates to the field of recursive filtering technology for nonlinear complex networks, and more specifically, to a recursive filtering method and system based on matrix decomposition in the case of sparse sensor measurements. Background Technology

[0002] This invention proposes a recursive filtering algorithm based on matrix factorization (MF) for the filtering problem of sensor networks in nonlinear complex network environments under low-duty-cycle scheduling (LDCS). The initial design intention is to solve the challenges and needs in the following technical background.

[0003] (1) Challenges of complex network environments: In complex network environments such as industrial IoT, smart grids and environmental monitoring, the widespread use of sensor networks has brought about many technical problems; High-dimensional sparse data: Sensor networks usually generate a large amount of high-dimensional data, and due to node failures, network packet loss or acquisition delays, the data is often incomplete or sparse. Dynamic noise and nonlinear environment: Data in complex networks is easily affected by external noise, and the network state and node characteristics are often highly nonlinear; Energy consumption constraints: Sensor nodes are usually powered by batteries, which will quickly deplete energy during frequent communication and computation, directly affecting the network's life cycle; Security threats: Distributed sensor networks are vulnerable to malicious attacks such as Denial of Service (DoS) attacks, affecting the stability of the system and the reliability of the data.

[0004] (2) Limitations of traditional filtering methods. Commonly used filtering techniques (such as unscented Kalman filtering and particle filtering) have the following shortcomings when dealing with complex network environments: high computational complexity, traditional algorithms need to process complete observation data, and it is difficult to effectively deal with large-scale high-dimensional data; poor adaptability to sparse data, lack of robustness to lost data, resulting in a decrease in filtering performance; neglect of energy consumption and security, traditional filtering algorithms usually focus on accuracy optimization and do not fully consider the energy efficiency and network security issues of sensor nodes.

[0005] (3) Technical advantages of matrix factorization and recursive filtering. Matrix factorization and recursive filtering methods have shown significant advantages in addressing the above problems. Matrix factorization can effectively handle sparse and incomplete data, extract low-dimensional features to reduce computational complexity, provide an efficient representation of data, and optimize filtering effects. Recursive filtering methods reduce computational overhead through stepwise updates, adapt to dynamic environmental changes, and are easily combined with techniques such as extended Kalman filtering to improve the ability to handle nonlinear systems.

[0006] (4) Combination of key optimization technologies. To further improve system performance, this invention combines a low duty cycle scheduling mechanism to extend the network lifetime by dynamically adjusting the activation time of sensor nodes. This adapts to different network load requirements and achieves a balance between energy efficiency and data accuracy. Based on Adam optimization and dynamic parameter adjustment, the Adam algorithm can dynamically adjust the learning rate, solving the problems of slow convergence speed and easy getting trapped in local optima in traditional gradient descent, and improving the system's adaptability to non-stationary noise environments. The DoS defense mechanism integrates anomaly detection and isolation functions, protecting the system from network attacks in real time, and dynamically adjusting defense strategies to cope with diverse security threats.

[0007] (5) Recursive error covariance matrix update. The filtering results are gradually optimized by recursively calculating the error covariance matrix. This method is particularly suitable for large-scale, dynamically changing sensor networks.

[0008] This invention combines matrix factorization technology, dynamic optimization algorithms, low duty cycle scheduling mechanisms, and DoS defense mechanisms to successfully address the high-precision, low-energy-consumption, and high-security requirements of filtering in complex network environments, demonstrating significant technical advantages and broad application prospects. Summary of the Invention

[0009] The technical problem to be solved by this invention is:

[0010] To address the issues of data loss, high energy consumption, multiple noise variations, and security in existing complex network environments for sensor networks.

[0011] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0012] This invention provides a recursive filtering method based on matrix decomposition for sparse sensor measurements, comprising the following steps:

[0013] S1. Sensor node deployment and parameter initialization: Deploy sensor nodes in the area to be monitored and build a network model to ensure that the nodes cover the entire area and configure the initial parameters of the nodes.

[0014] S2. Node data acquisition and low duty cycle scheduling: The sensor node acquires data according to the set sampling frequency and enters a low duty cycle mode according to the data acquisition load. It reduces power consumption through periodic wake-up and sleep to achieve dynamic scheduling.

[0015] S3. Data transmission and anomaly detection: The data collected in step S2 is transmitted to the gateway node. The system monitors the data flow in real time during the transmission process. When anomalies or missing data are detected, the system performs missing data processing to ensure data accuracy.

[0016] S4, DoS defense and data monitoring, monitors data traffic to determine if there is abnormal data. When a DoS attack is detected, the system isolates the abnormal node to ensure network security and normal data transmission.

[0017] S5. Matrix factorization data compensation: When data loss is detected, low-rank matrix factorization technology is used to complete the data. Missing data is filled by data from neighboring nodes to ensure the integrity of sensor network data.

[0018] S6. Adaptive state recursive filtering: Based on the state filter model, the sensor data is recursively filtered to make the state estimation accurate and adaptable to data fluctuations.

[0019] S7. Dynamic update of the error covariance matrix: The error covariance matrix is ​​updated based on the recursive filtering results to improve estimation accuracy and ensure the stability of the system in complex environments.

[0020] Furthermore, the recursive filtering method for the multi-robot localization system is applicable to discrete nonlinear complex network models with Gaussian noise or discrete nonlinear complex network models with multiple types of Gaussian noise.

[0021] Furthermore, when the recursive filtering method for the multi-robot localization system is applicable to a discrete nonlinear complex network model with Gaussian noise, it includes the following steps: S100, constructing a network model,

[0022]

[0023] in, and They represent time. Status input and measurement output before time transmission; x i,l+1 This represents the state input of the i-th sensor at time l+1; It is the external coupling configuration matrix of node i and node j. Let be the internal coupling strength matrix, where ; and These represent the process noise and measurement noise of node i at time l, respectively. and The zero-mean Gaussian white noise at node i at time l has the following covariance matrices: and And assume that at any time Down and They are independent of each other; It is a known nonlinear function. It is a known matrix;

[0024] S200 employs low duty cycle scheduling to manage data transmission, characterizing sensor measurements before and after transmission. The wireless sensor is configured as N nodes, and the measurement output before transmission is described as follows: In the formula, for Sensor nodes in The measurement output at time t; under low duty cycle scheduling, when the t... Each sensor node at time When in sleep mode, it will not send. ;make In the formula, for After the sensor node transmits The measurement output of time;

[0025] therefore, The update rules are as follows:

[0026]

[0027] in, and:

[0028]

[0029] in, For sensor nodes calculated based on the low duty cycle scheduling mechanism The periodicity of the function; It means l divided by The remainder;

[0030] S300, Adjust the duty cycle value to handle missing data, duty cycle for:

[0031]

[0032] In the formula, For sensor nodes Activation time within a cycle;

[0033] Based on formulas (1-3) and (1-4), the state of sensor node i, whether it is active or dormant, can be determined using the following method: If This indicates a sensor node. exist The sensor is always active and transmits data. , recorded as Otherwise, sensor nodes It is in sleep mode and does not send any data. and ;

[0034] In S400, MF and Adam optimization algorithms are used to predict untransmitted data, assuming that the sensor nodes use dimension... The latent vector representation, denoted as Time nodes are represented by dimensions. The vector representation of , denoted as Sensor measurement matrix Approximately By learning two matrices and This makes the original sensor measurement matrix With approximate matrix Minimize the difference between them; use the Adam optimization algorithm to iteratively update. and Parameters;

[0035] S500, Constructing a recursive filter:

[0036]

[0037] in, and Representing time respectively One step of prediction and status The filtered value; Is The filter gain is designed for the specific moment.

[0038] when At that time, node exist The filter is always active and receives data. The measurement output sent is ;when ,node exist When constantly in a dormant state, Combine the prediction vectors provided by MF ;

[0039] S600 introduces Theorem 3 and Theorem 4 to perform filter processing on nonlinear complex networks with multiplicative noise under low duty cycle scheduling when subjected to DoS attacks.

[0040] Furthermore, step S400 specifically includes,

[0041] S410, Define the loss function as... This is used to find the parameter values ​​that minimize the loss function;

[0042] The parameter updates for the S420 and Adam optimization algorithms are defined as follows:

[0043]

[0044] in, and Let i and l represent the first-moment estimates of the gradients at sensor node i and time node l, respectively. and Let represent the second-order moment estimates of the gradients at sensor node i and time node l, respectively. This represents the regularization parameter used to prevent the model from overfitting. and This represents the weights used to control the first and second moments of the gradient;

[0045] S430, Adaptive adjustment of the learning rate parameter, as detailed below.

[0046]

[0047] in, For learning rate, For adaptive learning rate parameters, This represents the number of iterations. These are the first-order moment weights and second-order moment weights after t iterations;

[0048] S440, Update the latent vector and :

[0049]

[0050] Here, ε is a small constant used to prevent division by zero errors; It is the updated sensor node vector; It is the updated time node vector;

[0051] S450, The predicted value of the data matrix that needs compensation is:

[0052]

[0053] in, The predicted vector value represents the measurement output:

[0054]

[0055] According to (1-2)-(1-13), the filter is defined in... The sensor measurement data received at any given time is:

[0056]

[0057] In formula (1-14), with Data This represents the sensor measurement data received by the filter, corresponding to the node. exist Instantaneous measurement value.

[0058] Furthermore, step S600 specifically includes,

[0059] Theorem 1: One-step prediction error covariance matrix and the filter error covariance matrix The following two recurrence equations are satisfied:

[0060]

[0061] in, This represents the external coupling configuration matrix of the i-th and z-th nodes of the sensor;

[0062] as well as,

[0063]

[0064] in, ;

[0065]

[0066] exist Expand using Taylor's formula ,get:

[0067]

[0068] In the formula , These are the higher-order components in the Jacobian matrix and Taylor series expression, respectively;

[0069] Will Represented as:

[0070]

[0071] in, To meet The time-varying matrix of unknown values, and Represent two known matrices;

[0072] The filtering error is obtained from formulas (1-1), (1-14), and (1-15):

[0073]

[0074] Theorem 2. Consider the one-step prediction error covariance matrix in formulas (1-16) and (1-17) respectively. and the filter error covariance matrix ;make and Let be a positive scalar; suppose the following two recursive matrix equations are:

[0075]

[0076] in, ; It is the upper bound matrix of the filtering error covariance matrix of the i-th sensor node at time l;

[0077] as well as,

[0078]

[0079] It has a positive definite solution. and initial conditions Then, the matrix yes The upper bound of the matrix yes The upper bound, namely:

[0080]

[0081] as well as,

[0082]

[0083] The filter gain matrix is ​​minimized using the upper bound matrix. The trace calculation, that is,

[0084]

[0085] in, .

[0086] Furthermore, when the recursive filtering method for the multi-robot localization system is applicable to a discrete nonlinear complex network model with various Gaussian noises, it includes the following steps: S100, constructing a network model,

[0087]

[0088] in, and These represent the pre-communication transmission steps. The state and measurement output at each moment; x i,l+1 This represents the state input of the i-th sensor at time l+1; Represents the external coupling configuration matrix; The inner coupling strength matrix is ​​represented as ; It is a zero-mean Gaussian random multiplicative noise variable with unit variance; and These are process noise and measurement noise, respectively. and Let be the zero-mean Gaussian white noise at node i at time l, and let their covariance matrices be respectively. and Assuming , and In any Time is irrelevant; It is a known nonlinear function; and All are known matrices;

[0089] S200 employs low duty cycle scheduling to manage data transmission, dividing the sensor into... Each node, the measurement output before transmission is described as follows: In the formula Indicates in From time to time Measurement output obtained by sensor nodes; under low duty cycle scheduling Not in the sensor exist It is sent while the device is in a dormant state.

[0090] S300, Order In the formula For the first The sensor node at the ... The measurement output of time; The update rules are as follows:

[0091]

[0092] S400, where, Represents random variables Describe whether a DoS attack has occurred, considering each node... The probability of being attacked is equal; for And it follows a Bernoulli distribution, with the probability distribution of its random variable being:

[0093]

[0094] in, It is a known constant, and ,

[0095]

[0096] Duty cycle for:

[0097]

[0098] In the formula, For sensors Activation time within a cycle;

[0099] S500, the measurement output received by the filter, constructed according to formulas (2-2)-(2-5), is defined as follows: , ;

[0100] S600, construct the following recursive filter:

[0101]

[0102] in, and Representing time respectively One step of prediction and status The filtered value; Is The filter gain is designed for the specific moment.

[0103] S700 introduces Theorem 3 and Theorem 4 to perform filter processing on nonlinear complex networks with multiplicative noise under low duty cycle scheduling when subjected to DoS attacks.

[0104] Further, step S700 specifically includes,

[0105] Theorem 3: One-step prediction error covariance matrix and the filter error covariance matrix The following two recurrence equations are satisfied:

[0106]

[0107] and,

[0108]

[0109] in, ;

[0110]

[0111] exist Expand using Taylor's formula ,get:

[0112]

[0113] In the formula, , These are the higher-order components in the Jacobian matrix and Taylor series expression, respectively;

[0114] Represented as:

[0115]

[0116] in, To meet The unknown time-varying matrix, and Represent two known matrices;

[0117] The filtering error can be obtained from (2-1) and (2-6):

[0118]

[0119] The filter gain is determined by minimizing the trace of this upper bound;

[0120] Theorem 4: Consider the one-step prediction error covariance matrix in formulas (2-7) and (2-8) respectively. and the filter error covariance matrix ;make and Let be a positive scalar; suppose the following two recursive matrix equations are:

[0121]

[0122] in, ; It is the upper bound matrix of the filtering error covariance matrix of the i-th sensor node at time l;

[0123] as well as,

[0124]

[0125] It has a positive definite solution. and initial conditions Then, the matrix yes The upper bound of the matrix yes The upper bound, that is,

[0126]

[0127] as well as,

[0128]

[0129] The filter gain matrix is ​​minimized using the upper bound matrix. The trace calculation, that is,

[0130]

[0131] in, .

[0132] This invention provides a recursive filtering system based on matrix decomposition for sparse sensor measurements. The system has a program module corresponding to the above steps, and executes the steps in the above-described recursive filtering method based on matrix decomposition for sparse sensor measurements during runtime.

[0133] The present invention provides a computer-readable storage medium storing a computer program configured to, when called by a processor, implement the steps of a recursive filtering method based on matrix factorization in the case of sparse sensor measurements.

[0134] Compared with the prior art, the beneficial effects of the present invention are:

[0135] To address the high energy consumption caused by frequent wake-ups of existing sensor nodes, this invention controls the activation time of sensor nodes through a low duty cycle scheduling mechanism (LDCS), ensuring that sensor nodes remain in a dormant state during inactive periods. This reduces unnecessary data duplication, effectively lowers sensor energy consumption, and extends sensor lifespan. Furthermore, existing methods such as deep learning and neural network algorithms require model training when handling high data loss rates, resulting in poor real-time performance and high hardware requirements and computational demands. This invention, combining matrix factorization and its optimization algorithms, effectively reduces computational resource consumption while ensuring data reconstruction accuracy. For example, with a 60% data loss rate at some nodes, the error is less than 5%, while traditional methods have errors as high as 30%. This invention uses matrix factorization technology to process sparse data, improving the ability to handle incomplete or missing data and reducing computational burden. By dynamically adjusting the parameters of the Adam optimization algorithm, this invention adapts to different network noise environments and nonlinear data characteristics, thereby improving data accuracy. Through a DoS defense mechanism, this invention uses a random variable model to identify abnormal fluctuations in the data, monitors and isolates potential attack data in real time, and prevents malicious data from affecting the overall system performance. Furthermore, this invention addresses the problem of gradual error accumulation in traditional Kalman filtering when dealing with continuous data loss, leading to progressively increasing errors and potentially causing system divergence or even collapse. By compensating for sparse data and deriving the optimal filter gain matrix from the upper bound of the filter error covariance matrix, this invention effectively avoids the accumulation of system filtering errors, improves system robustness, and also possesses the advantage of real-time calculation of recursive filtering. This achieves gradual data updates and accurate filtering, ensuring the system can still work effectively when faced with sparse or lost data. Attached Figure Description

[0136] Figure 1 This is a flowchart of a recursive filtering method based on matrix decomposition in the case of sparse sensor measurements in an embodiment of the present invention;

[0137] Figure 2 This is a framework diagram of the recursive filtering method based on matrix decomposition in the case of sparse sensor measurements in an embodiment of the present invention;

[0138] Figure 3 This is a diagram showing the relationship between duty cycle, activation time, and sleep time in an embodiment of the present invention.

[0139] Figures 4-10 This is a comparison chart of the estimation performance and error curves of the algorithm and the traditional algorithm at different duty cycles in this embodiment of the invention;

[0140] Figure 11-14 This is a comparison of the estimation performance and error curves of the algorithm and the traditional algorithm under the condition that duty cycle and DoS attack coexist in the embodiment of the present invention;

[0141] Figure 15-18 This is a comparison chart of the estimation performance and error curves of the algorithm and the traditional algorithm during a DoS attack in an embodiment of the present invention;

[0142] Figures 19-20 This is a diagram showing the triggering state of the system output values ​​of each node under a DoS attack at different times in an embodiment of the present invention. Detailed Implementation

[0143] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0144] Some parameters of this invention are defined as follows: express Vie Euclidean space. The modulo function. express Divide by The remainder. Represents an identity matrix of appropriate dimensions. The norm of a matrix is ​​represented. and Represent matrices respectively The inverse matrix and transpose matrix. express The expected value. Represents the trace of a matrix.

[0145] Specific Implementation Plan 1: Combining Figures 1 to 3 As shown, this invention provides a recursive filtering method based on matrix decomposition for sparse sensor measurements, comprising the following steps:

[0146] S1. Sensor node deployment and parameter initialization, and network model construction: Deploy sensor nodes in the area to be monitored to ensure that the nodes cover the entire area, and configure the initial parameters of the nodes.

[0147] S2. Node Data Acquisition and Low Duty Cycle Scheduling: Sensor nodes acquire data according to the set sampling frequency and enter a low duty cycle mode based on the data acquisition load. Dynamic scheduling is achieved by periodically waking up and sleeping to reduce power consumption.

[0148] By setting an adjustable ratio of activation time to cycle time, and combining this with changes in network topology, the activation frequency of nodes can be adjusted, thereby effectively reducing the overall energy consumption of the system and extending the lifespan of the sensor network.

[0149] The LDCS time interval can be dynamically matched with the network load or the real-time status of the system, so that the activation time of each node is as appropriate as possible to its importance and task priority, thereby achieving a balance between energy efficiency and data accuracy.

[0150] S3. Data transmission and anomaly detection: The collected data is transmitted to the gateway node via wireless communication. The system monitors the data flow in real time during the transmission process. If abnormal data or missing data is detected (abnormal data refers to values ​​that are outside the range, as well as high-frequency noise values, values ​​that are attenuated over long distances, etc.), anomaly processing is performed immediately (for abnormal data, if the abnormal data is missing, the same value as the missing value is compensated) to ensure data integrity.

[0151] S4. DoS Defense and Data Monitoring: Monitor data traffic to determine if there is any abnormal data (DoS attacks are considered to occur randomly with a certain probability. When an attack occurs, the attacker will send abnormal or false data. This attack data will be missing or deleted, which is passive data loss and will be treated as missing data later). When a possible DoS attack is detected, the system isolates abnormal nodes to ensure network security and normal data transmission.

[0152] DoS defense mechanisms include:

[0153] S41, Anomaly Detection Module: Uses random probability statistical detection methods to monitor node data streams in real time and determine whether there is abnormal data based on preset thresholds;

[0154] S42, Abnormal Data Isolation Module: Once abnormal data is detected, data shielding measures are immediately taken to prevent attack data from interfering with the network;

[0155] S43. Adaptive adjustment of defense strategy: Dynamically adjust the DoS defense mechanism according to the changing patterns of network attacks, so that the system can automatically respond to attacks and interference;

[0156] S5. Matrix factorization data compensation: When data loss is detected, low-rank matrix factorization is used to complete the data. Missing data is filled in by data from neighboring nodes to ensure the integrity of sensor network data. The original sparse matrix is ​​decomposed into a low-rank matrix. This decomposition optimizes the data representation, reduces the computational complexity of the system, and maintains the accuracy of filtering.

[0157] The matrix factorization technique can also be combined with compressed sensing to perform sparse coding and signal recovery on the data collected in the network, thereby further improving the efficiency and accuracy of data processing. Especially in scenarios where there is a lot of redundant information in sensor data, compressed sensing can greatly reduce the amount of computation.

[0158] S6. Adaptive State Recursive Filtering: Based on the state filter model, sensor data is recursively filtered. The Adam optimization algorithm adaptively adjusts the parameters to make the state estimation accurate and adaptable to data fluctuations. The Adam optimization algorithm adaptively adjusts the parameters by calculating the adaptive learning rate and the first / second moment estimates to avoid the traditional gradient descent algorithm from having too slow a convergence speed or getting stuck in local minima in non-stationary environments, thereby accelerating the convergence process of the system.

[0159] Specifically, including,

[0160] S61. Initialization phase: Set the initial filter error covariance matrix using prior knowledge or estimated initial state values;

[0161] S62. Prediction Stage: Predict the current state using a recursive filtering algorithm and estimate the error covariance matrix;

[0162] S63. Update phase: Based on the new observation data and prediction results, the filter is corrected and the covariance matrix is ​​updated to accurately estimate the dynamic state of the system.

[0163] The dynamic parameter adjustment based on the Adam optimization algorithm adopts the stochastic gradient descent algorithm to optimize the system parameters after each data update, so that the system can quickly adapt to and maintain high accuracy when facing sudden environmental changes (such as node damage or network topology changes).

[0164] S7. Dynamic update of recursive filtering error covariance matrix: The error covariance matrix is ​​updated based on the recursive filtering results to improve estimation accuracy and ensure system stability in complex environments. The recursive filtering error covariance matrix update automatically calculates and updates the error covariance matrix by utilizing the extended Kalman filtering method, ensuring that the filtering error is continuously optimized over time, thereby maintaining high filtering accuracy when facing nonlinear systems and large-scale sensor network nodes.

[0165] Specific Implementation Plan Two: Consider a class of discrete nonlinear complex network models with Gaussian noise, including the following steps:

[0166] Building the network model:

[0167]

[0168] in, and They represent time. Status input and measurement output before time transmission; x i,l+1 This represents the state input of the i-th sensor (same meaning as a node) at time (same meaning as moment) l+1; It is the external coupling configuration matrix of node i and node j. Let be the internal coupling strength matrix, where ; and These represent the process noise and measurement noise of node i at time l, respectively. and The noise at node i at time l is zero-mean Gaussian white noise, and the covariance matrices are respectively and And assume that at any time Down and They are independent of each other; It is a known nonlinear function. It is a known matrix;

[0169] To achieve accurate filtering under a low duty cycle scheduling mechanism, it is necessary to characterize the sensor measurements before and after transmission. Changes in the measurements after network transmission directly affect the completeness and accuracy of available measurements, thus significantly impacting the system's filtering performance. Specifically, considering the spatial distribution of sensors in practical engineering applications, the wireless sensor (similar to the sensor in this invention) can be divided into N nodes. For ease of theoretical analysis, the measurement output before transmission can be described as... In the formula for Sensor nodes in The measurement output at time; it should be noted that under low duty cycle scheduling, when the... Each sensor node at time When in sleep mode, it will not send. ;make In the formula for After the sensor node transmits The measurement output at time; therefore, The update rules are as follows:

[0170]

[0171] in, and:

[0172]

[0173] in, For sensor nodes calculated based on the low duty cycle scheduling mechanism The cycle time (cycle time refers to the minimum time required for a sensor to periodically activate and then remain dormant); the function mod express Divide by The remainder;

[0174] Duty cycle for:

[0175]

[0176] In the formula, For sensor nodes Activation time within a cycle;

[0177] Based on formulas (1-3) and (1-4), the state of sensor node i, whether it is active or dormant, can be determined using the following method: If This indicates a sensor node. exist If the sensor is constantly active, then the data it transmits... Therefore, it is recorded as Otherwise, sensor nodes It is in sleep mode and does not send any data. and Activation time hibernation time and cycle time The interrelationship between them, such as Figure 3 As shown; clearly, the duty cycle makes a significant contribution to reducing sensor energy consumption; a lower duty cycle results in greater sensor energy savings; however, it also leads to sparser data received by the filter; therefore, in this scheduling process, the filtering performance degrades due to the sparse received data.

[0178] Traditionally, a zero-order holder (ZOH) is used to fill in missing data to establish an effective and reliable strategy for handling untransmitted data. However, the ZOH method is not suitable for the current research context. If ZOH is used to handle untransmitted data, the system will replace all untransmitted sensor data with the last set of data transmitted in the previous activation cycle. This approach has the following problems: the data received by the filter remains unchanged for a period of time, lacking real-time updates. Obviously, this data freeze phenomenon not only affects the system's responsiveness to real-time environmental changes but also affects the system's operation, leading to a significant decrease in system performance and even instability in some cases. Therefore, in order to maintain the real-time performance and stability of the system, it is imperative to seek a more reasonable strategy.

[0179] In wireless sensor networks, data transmission may be incomplete or even interrupted due to node sleep or communication. To address this issue, Matrix Factorization (MF) and Adam optimization algorithms are used to predict untransmitted data. This is not only because this strategy has unique advantages in handling sparse and high-dimensional data, but also because it can meet the needs of real-time computing with minimal hardware requirements. MF is a commonly used machine learning algorithm, also known as matrix completion or matrix reconstruction. Its principle is to approximate the original matrix by decomposing a large matrix into the product of two low-rank matrices. By learning the parameters of these two matrices, important sensor data features can be extracted for tasks such as prediction, interpolation, or anomaly detection. The prediction process of MF includes: assuming that sensor nodes can be represented by a matrix with dimension... The latent vector representation, denoted as Similarly, time points can be represented by dimensions. The vector representation of , denoted as Therefore, the sensor measurement matrix It can be approximated as The goal is to learn two matrices. and This makes the original sensor measurement matrix With approximate matrix The goal is to minimize the difference between the two values; to achieve this, adaptive moment estimation, also known as the Adam optimization algorithm, is employed, iteratively updating... and The parameters include:

[0180] 1) First, define the loss function as... The goal is to find parameter values ​​that minimize the loss function;

[0181] 2) The parameter update of the Adam optimization algorithm is defined as follows:

[0182]

[0183] in, and Let i and l represent the first-moment estimates of the gradients at sensor node i and time node l, respectively. and Let represent the second-order moment estimates of the gradients at sensor node i and time node l, respectively. This represents the regularization parameter used to prevent the model from overfitting. and This represents the weights used to control the first and second moments of the gradient;

[0184] 3) Adaptive adjustment of the learning rate parameter, as detailed below.

[0185]

[0186] in, For learning rate, For adaptive learning rate parameters, This represents the number of iterations. These are the first-order moment weights and second-order moment weights after t iterations;

[0187] 4) Update the latent vector and :

[0188]

[0189] Here, ε is a small constant used to prevent division by zero errors; It is the updated sensor node vector; It is the updated time node vector;

[0190] 5) The predicted value of the final data matrix that needs compensation is:

[0191]

[0192] in, This represents the predicted vector value of the measurement output, which is calculated using the following formula:

[0193]

[0194] According to formulas (1-2) to (1-13), the filter is defined in... The sensor measurement data received at any given time is:

[0195]

[0196] In formula (1-14), with Data This represents the sensor measurement data received by the filter, which corresponds to the node. exist The instantaneous measurement value is then used to construct a recursive filter with the following structure:

[0197]

[0198] in, and Representing time respectively One step of prediction and status The filtered value; Is The filter gain designed for each time step; when the left side of | is l, it represents time l; when the left side of | is l+1, it represents time l+1, as shown below. , which are the parameters at time l+1;

[0199] The recursive filter (1-15) receives The composition, as shown in formula (1-14), depends on the transmission state of the sensor nodes, as represented by formula (1-13); when This means the node exist The filter is always active and receives data. Only the measurement output is sent. ;when , indicating the node In real time When in hibernation, Combine the prediction vectors provided by MF It is used to compensate for the measurement values ​​that the sensor loses due to the sleep state, thereby ensuring the real-time and completeness of the sensor data, while also taking into account the low power consumption characteristics of the sensor.

[0200] Subsequently, a filtering strategy is studied in depth through Theorem 1.1 and Theorem 1.2, and a rigorous theoretical analysis is conducted to evaluate its boundedness and stability, thereby ensuring its effectiveness and reliability.

[0201] Theorem 1.1: One-step prediction error covariance matrix and the filter error covariance matrix The following two recurrence equations are satisfied:

[0202]

[0203] in, Represents the external coupling configuration matrix of the i-th and z-th nodes of the sensor; and,

[0204]

[0205] in, ;

[0206] The proof is as follows: Based on formulas (1-1) and (1-15), we can derive the following:

[0207]

[0208] exist Expand using Taylor's formula ,get:

[0209]

[0210] In the formula , These are the higher-order components in the Jacobian matrix and Taylor series expression, respectively;

[0211] Can Represented as:

[0212]

[0213] in, To meet The time-varying matrix of unknown values, and Let (1-1) represent two known matrices; the one-step prediction error covariance matrix can be easily obtained from (1-1). As shown in (1-16);

[0214] The filtering error can be obtained using formulas (1-1), (1-14), and (1-15):

[0215]

[0216] Therefore, the filter error covariance matrix This can be derived as formula (1-17);

[0217] It can be clearly seen from formula (1-16) that, due to the introduction of nonlinearity, the one-step prediction error covariance matrix... It includes uncertainties; therefore, the filter error covariance matrix is ​​obtained. The exact value of the filter gain is nearly impossible to find, which in turn hinders the calculation of the filter gain; this section will address this problem by considering the upper bound of the filter error covariance matrix and then determining the filter gain by minimizing the trace of that upper bound.

[0218] Theorem 1.2: Consider the one-step prediction error covariance matrix in Equations (1-16) and (1-17) respectively. and the filter error covariance matrix ;make and It is a positive scalar; suppose the following two recursive matrix equations are:

[0219]

[0220] in, ; It is the upper bound matrix of the filter error covariance matrix of the i-th sensor node at time l; and,

[0221]

[0222] It has a positive definite solution. and initial conditions Then, the matrix yes The upper bound of the matrix yes The upper bound, namely:

[0223]

[0224] as well as,

[0225]

[0226] The filter gain matrix is ​​minimized using the upper bound matrix. The trace calculation, that is,

[0227]

[0228] in,

[0229] The subsequent lemmas will be used to establish the main results of this invention;

[0230] Lemma 1.1: For any two vectors The following inequalities hold:

[0231]

[0232] in, It is a constant scalar;

[0233] Lemma 1.2: For a given matrix and ,in satisfy If there exists a symmetric matrix and an additional variable satisfy Then the following matrix inequalities hold:

[0234]

[0235] Lemma 1.3: For any matrix ,if Then the following inequalities hold:

[0236]

[0237] in It is a square matrix with a known dimension;

[0238] Lemma 1.4: Assumptions .

[0239] If it exists This makes the following equation true:

[0240]

[0241] Where Z is a real symmetric matrix, X l (Z) represents a real symmetric discrete difference equation, equivalent to the error covariance matrix equation; Y l (Z) also represents a real symmetric discrete difference equation;

[0242] Then the solution to the following difference equation :

[0243]

[0244] Where W represents a real symmetric matrix; R0 and S0 represent the initial conditions, i.e., initial values, of the discrete difference equation;

[0245] satisfy ;

[0246] Lemma 1.5: For a given deterministic matrix The following equation holds true:

[0247]

[0248] in Represents the elements in the matrix, that is, the specific values ​​in the matrix;

[0249] Lemma 1.6: Suppose that for any and There exists a positive scalar , , , , , , , , , , and It meets the following conditions:

[0250]

[0251] Then the difference equation and It is bounded, that is, , ,

[0252] in, ;

[0253] The proof is as follows: based on formula (1-22) and the assumptions, we can obtain... The upper bound; substituting formula (1-26) into formula (1-23) yields:

[0254] Proof complete;

[0255] To prove Theorem 1.2, mathematical induction is used here; consider the initial conditions. And assume that for any for We will continue to prove that the inequality still holds in subsequent steps. Specifically, It also holds true;

[0256] First, for the first term on the right-hand side of equation (1-16), using Lemma 1.2, we can obtain...

[0257]

[0258] For the second term on the right-hand side of equation (1-16), by applying Lemma 1.1, we have:

[0259]

[0260] here Similarly, it can be deduced that...

[0261]

[0262] From formulas (1-33), (1-34), and (1-35), we can derive:

[0263]

[0264] Next, the second item on the right side of (1-17) can be handled in a similar manner; similarly, it can be derived from Lemma 1.1:

[0265]

[0266] Substituting (1-37) into (1-17) yields:

[0267]

[0268] Furthermore, formula (1-23) can be easily derived from inequality (1-24) and lemma 1.4, and combined with the initial conditions. and assumptions Now, find the matrix. traces Since the partial derivatives are equal to zero, we can obtain:

[0269]

[0270] Therefore, it was observed The term is invertible, and the filter gain matrix can be obtained as shown in formula (1-26); now the proof of Theorem 1.2 is complete.

[0271] Simulation Experiment 1

[0272] Next, the effectiveness of the proposed filtering algorithm will be verified by a simulation experiment; assuming a system with 8 nodes (1-1), the selected parameters are as follows, where the matrix... and nonlinear functions Set as:

[0273] ,

[0274] in, This represents the first component of the state x of the i-th sensor at time l; Let x represent the second component of the state x of the i-th sensor at time l; for example, if a sensor can measure both position and velocity, the position is the first component and the velocity is the second component.

[0275] Other parameters are , ,in, That is, a unitary two-dimensional matrix; measuring noise and process noise It is zero-mean white noise with covariance of , , , , , , and .here This represents the state vector. The initial state is... , , , .also, , , , , , , , , as well as Considering the transmission of measurements under LDCS in a wireless network, when the latent vector dimension... Learning rate Regularization parameters The filtering results of the method proposed in this invention were observed.

[0276] Observe the duty cycle from arrive The filtering effect at that time, the result is as follows Figure 4 - Figure 9 As shown. From Figure 10 It can be seen that the filtering performance of traditional ZOH-based methods is severely degraded due to the sparsity of measurements under LDCS. To improve filtering performance, a recursive filtering algorithm based on MF was designed, which improves the accuracy and reliability of the filter. Compared with the traditional ZOH-based method, these figures show that the algorithm of this invention can still achieve satisfactory filtering performance under low duty cycles. Simulation results ultimately show that combining the recursive filtering algorithm with the MF designed in this invention can accurately filter the state of system (1-1) under sparse data. Through comparative analysis with the traditional ZOH-based method, this algorithm ensures the filtering performance of the system under LDCS scheduling. The research results have strong theoretical and engineering application value.

[0277] Specific implementation plan three: Consider a class of discrete nonlinear complex network models with various types of Gaussian noise, including the following steps:

[0278] Building the network model:

[0279]

[0280] in, and These represent the pre-communication transmission steps. The status input and measurement output at any given time; Represents the external coupling configuration matrix; The inner coupling strength matrix is ​​represented as ; It is a zero-mean Gaussian random multiplicative noise variable with unit variance; and These are process noise and measurement noise, respectively. and The noise is zero-mean Gaussian white noise, and the covariance matrices are respectively and Assuming , and In any Time is irrelevant; It is a known nonlinear function; and All are known matrices;

[0281] Considering that data in a wireless sensor network system is transmitted by sensors through a communication network, LDCS (Low-Level Sensor Control) is used to manage data transmission in order to save energy and extend sensor lifespan. In this method, the sensor's operating state alternates between sleep and active states, influenced by LDCS. Based on the spatial distribution of sensors in practical engineering, it is proposed that wireless sensors can be divided into... There are several nodes; for ease of theoretical analysis, the measurement output before transmission can be described as follows: In the formula Indicates in From time to time The measurement output obtained by the sensor node; it should be noted that under LDCS, Will not be in the sensor exist It is sent while the device is in a dormant state.

[0282] make In the formula For the first The sensor node at the ... The measurement output at time; therefore, The update rules are as follows:

[0283]

[0284] in Represents random variables Describe whether a DoS attack has occurred, considering each node... The probability of being attacked is equal; for And it follows a Bernoulli distribution, with the probability distribution of its random variable being:

[0285]

[0286] in, It is a known constant, and ,

[0287]

[0288] in, For calculation based on LDCS Sensor node cycle time, duty cycle for:

[0289]

[0290] In the formula, For sensors Activation time within a cycle;

[0291] Based on formulas (2-2)-(2-5) and (1-5)-(1-13), the measurement output received by the filter is defined as follows: , ;

[0292] Then construct a recursive filter with the following structure.

[0293]

[0294] in and Representing time respectively One step of prediction and status The filtered value; Is The filter gain is designed for the specific moment.

[0295] This section aims to develop a feasible sparse data processing method for complex nonlinear networks with multiplicative noise under LDCS when subjected to DoS attacks. Subsequently, a filtering strategy will be developed and rigorously analyzed to evaluate its bounded stability, thereby ensuring its effectiveness and reliability.

[0296] Theorem 2.1: One-step prediction error covariance matrix and the filter error covariance matrix The following two recurrence equations are satisfied:

[0297]

[0298] as well as,

[0299]

[0300] in, ;

[0301] The proof is as follows. Based on formulas (2-1) and (2-6), we can derive:

[0302]

[0303] exist Expand using Taylor's formula ,get:

[0304]

[0305] In the formula, , These are the higher-order components in the Jacobian matrix and Taylor series expression, respectively;

[0306] Can Represented as:

[0307]

[0308] in, To meet The unknown time-varying matrix, and Let (1-1) represent two known matrices; the one-step prediction error covariance matrix can be easily obtained from (1-1). As shown in (2-7);

[0309] The filtering error can be obtained from formulas (2-1) and (2-6):

[0310]

[0311] Therefore, the filter error covariance matrix This can be derived as formula (2-8);

[0312] It can be clearly seen from formula (2-7) that, due to the introduction of nonlinearity, the one-step prediction error covariance matrix... It includes uncertainties; therefore, the filter error covariance matrix is ​​obtained. The exact value of the filter gain is nearly impossible to find, which in turn hinders the calculation of the filter gain; this section will address this problem by considering the upper bound of the filter error covariance matrix and then determining the filter gain by minimizing the trace of that upper bound.

[0313] Theorem 2.2: Consider the one-step prediction error covariance matrix in Equations (2-7) and (2-8) respectively. and the filter error covariance matrix ;make and Let be a positive scalar; suppose the following two recursive matrix equations are:

[0314]

[0315] as well as,

[0316]

[0317] It has a positive definite solution. and initial conditions Then, the matrix yes The upper bound of the matrix yes The upper bound, that is,

[0318]

[0319] as well as,

[0320]

[0321] The filter gain matrix is ​​minimized using the upper bound matrix. The trace calculation, that is,

[0322]

[0323] in,

[0324]

[0325] To prove Theorem 2.2, mathematical induction is used here; consider the initial conditions. And assume that for any for We continue to prove that the inequality still holds in subsequent steps, specifically, It also holds true;

[0326] First, for the first term on the right-hand side of equation (2-7), using Lemma 2, we can derive:

[0327]

[0328] For the second term on the right-hand side of formula (2-7), using Lemma 1.1, we can derive:

[0329]

[0330] in, Similarly, we can further obtain:

[0331]

[0332] For the fourth term on the right-hand side of equation (2-7), using Lemma 1.1, we can see that:

[0333]

[0334] From formulas (2-18), (2-19), and (2-20), we can obtain:

[0335]

[0336] Next, we will use a similar method to handle the second item on the right side of equation (2-8); again, it can be obtained from Lemma 1.1:

[0337]

[0338] Substituting formula (2-23) into formula (2-8) yields the following result:

[0339]

[0340] Furthermore, formula (2-14) can be easily derived from inequality (2-16) and lemma 1.4, and combined with the initial conditions. and assumptions Now, find the matrix. traces The partial derivative of is equal to zero, which gives us...

[0341]

[0342] Therefore, the filter gain matrix can be obtained as shown in formula (2-17), and It is reversible; the proof of Theorem 2.2 is now complete.

[0343] Assuming for any and There exists a positive scalar , , , , , , , , , , , , and The following conditions must be met.

[0344] ,

[0345] Then the difference equation and It is bounded, that is, , ,

[0346] in,

[0347] ,

[0348] The proof is as follows: based on formula (2-13) and the assumptions, the following can be derived. The upper bound; substituting formula (2-17) into formula (2-14) yields...

[0349] Difference equations and The boundedness proof is complete.

[0350] Simulation Experiment 2

[0351] In modern networked intelligent systems, robotics has become a crucial research area, especially in indoor environments where robot localization technology has wide applications in smart homes, logistics, search and rescue, and other fields. However, traditional localization methods are often limited by environmental factors, sensor accuracy, and algorithm implementation due to the complexity of indoor environments. Therefore, designing an efficient and robust localization system, particularly for the localization of multiple mobile robots in complex network environments, has become an important research topic. This simulation focuses on the localization problem of multiple mobile robots in an indoor environment, exploring a multi-robot localization system based on a complex network model from an algorithmic perspective, and evaluating its performance, stability, and accuracy through simulation analysis.

[0352] The following is a simulation analysis of the indoor localization problem of four mobile robots in a complex network with eight sensor measurement nodes. Mobile robots often encounter state constraints in practical applications, such as position and orientation limitations. Furthermore, based on the kinematic equations of the mobile robots, consider the following system:

[0353]

[0354] here, Zero-mean, unit-variance multiplicative noise The nonlinear terms of the system are set as follows:

[0355] , ,

[0356] Noise measurement and process noise Both are zero-mean Gaussian white noise, and their covariance matrices are respectively , , , , , , , .in Indicates time Time The state vectors of the robot in the horizontal and vertical positions. and Indicates direction and number A robot in time The velocity vector at time.

[0357] Initial parameters are , , Other parameters are selected as follows: , , , , , $, , , , , , Assume the probability of a DoS attack is 0.2. Considering the transmission of measurements via an LDCS-based wireless network, when the potential vector dimension is... The learning rate is The regularization parameter is At that time, the filtering effects of different methods were observed.

[0358] Figure 11-14 The system's state filter value, sensor output value, and root mean square error (RMSE) of the state filter are shown under the condition that duty cycle and DoS attack coexist. Figure 15-18 The display shows only the system state filter values, sensor output values, and RMSE during a DoS attack. Figure 19 and Figure 20 The paper demonstrates the trigger states of the system's sensor output values ​​at different times under a DoS attack. The comparison shows that the designed MF-based data compensation algorithm is significantly superior to the traditional zero-order hold algorithm. The algorithm designed in this invention achieves high-precision calculation of the filter under different duty cycles, balancing the advantages of system energy saving and high filtering performance.

[0359] As can be seen from the above analysis, the sparse data-driven recursive filtering algorithm of this application is applicable to networked systems under different conditions. It considers the influence of phenomena such as duty cycle scheduling, DoS attacks, nonlinearity, complex networks, multiplicative noise, and additive noise on system parameters, constructs a discrete time-varying model of the system and a filter model, and obtains the filtering gain by minimizing the trace of the filtering error covariance matrix. Furthermore, the boundedness of the filtering error is analyzed. Compared with traditional recursive filtering algorithms, this invention is applicable to common networked systems in practical applications and has general applicability.

[0360] The other combinations and connections in this implementation scheme are the same as in Specific Implementation Scheme 1.

[0361] While the present invention has been disclosed above, its scope of protection is not limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention, and all such changes and modifications will fall within the scope of protection of the present invention.

Claims

1. A recursive filtering method based on matrix decomposition for sparse sensor measurements, characterized in that, Includes the following steps: S1. Sensor node deployment and parameter initialization: Deploy sensor nodes in the area to be monitored and build a network model to ensure that the nodes cover the entire area and configure the initial parameters of the nodes. S2. Node data acquisition and low duty cycle scheduling: The sensor node acquires data according to the set sampling frequency and enters a low duty cycle mode according to the data acquisition load. It reduces power consumption and achieves dynamic scheduling through periodic wake-up and sleep. S3. Data transmission and anomaly detection: The data collected in step S2 is transmitted to the gateway node. The system monitors the data flow in real time during the transmission process. When anomalies or missing data are detected, the system performs missing data processing to ensure data accuracy. S4, DoS defense and data monitoring, monitors data traffic to determine if there is abnormal data. When a DoS attack is detected, the system isolates the abnormal node to ensure network security and normal data transmission. S5. Matrix factorization data compensation: When data loss is detected, low-rank matrix factorization technology is used to complete the data. Missing data is filled by data from neighboring nodes to ensure the integrity of sensor network data. S6. Adaptive state recursive filtering: Based on the state filter model, the sensor data is recursively filtered to make the state estimation accurate and adaptable to data fluctuations. S7. Dynamic update of the error covariance matrix: The error covariance matrix is ​​updated based on the recursive filtering results to improve estimation accuracy and ensure the stability of the system in complex environments.

2. The recursive filtering method based on matrix decomposition for sparse sensor measurements according to claim 1, characterized in that: The recursive filtering method for the multi-robot localization system is applicable to discrete nonlinear complex network models with Gaussian noise or discrete nonlinear complex network models with multiple types of Gaussian noise.

3. The recursive filtering method based on matrix decomposition for sparse sensor measurements according to claim 2, characterized in that: When the recursive filtering method of the multi-robot localization system is applicable to a discrete nonlinear complex network model with Gaussian noise, it includes the following steps: S100, constructing a network model, in, and They represent time. Status input and measurement output before time transmission; x i,l+1 This represents the state input of the i-th sensor at time l+1; It is the external coupling configuration matrix of node i and node j. Let be the internal coupling strength matrix, where ; and These represent the process noise and measurement noise of node i at time l, respectively. and The zero-mean Gaussian white noise at node i at time l has the following covariance matrices: and And assume that at any time Down and They are independent of each other; It is a known nonlinear function. It is a known matrix; S200 employs low duty cycle scheduling to manage data transmission, characterizing sensor measurements before and after transmission. The wireless sensor is configured as N nodes, and the measurement output before transmission is described as follows: In the formula, for Sensor nodes in The measurement output at time t; under low duty cycle scheduling, when the t... Each sensor node at time When in sleep mode, it will not send. ;make In the formula, for After the sensor node transmits The measurement output of time; therefore, The update rules are as follows: in, and: in, For sensor nodes calculated based on the low duty cycle scheduling mechanism The periodicity of the function; It means l divided by The remainder; S300, Adjust the duty cycle value to handle missing data, duty cycle for: In the formula, For sensor nodes Activation time within a cycle; Based on formulas (1-3) and (1-4), the state of sensor node i, whether it is active or dormant, can be determined using the following method: If This indicates a sensor node. exist The sensor is always active and transmits data. , recorded as Otherwise, sensor nodes It is in sleep mode and does not send any data. and ; In S400, MF and Adam optimization algorithms are used to predict untransmitted data, assuming that the sensor nodes use dimension... The latent vector representation, denoted as Time nodes are represented by dimensions. The vector representation of , denoted as Sensor measurement matrix Approximately By learning two matrices and This makes the original sensor measurement matrix With approximate matrix Minimize the difference between them; use the Adam optimization algorithm for iterative updates. and Parameters; S500, Constructing a recursive filter: in, and Representing time respectively One step of prediction and status The filtered value; Is The filter gain is designed for the specific moment. when At that time, node exist The filter is always active and receives data. The measurement output sent is ;when ,node exist When constantly in a dormant state, Combine the prediction vectors provided by MF ; S600 performs filter processing on nonlinear complex networks with multiplicative noise under low duty cycle scheduling when subjected to DoS attacks.

4. The recursive filtering method based on matrix decomposition for sparse sensor measurements according to claim 3, characterized in that: Step S400 specifically includes, S410. Define the loss function as... This is used to find the parameter values ​​that minimize the loss function; The parameter updates for the S420 and Adam optimization algorithms are defined as follows: in, and Let i and l represent the first-moment estimates of the gradients at sensor node i and time node l, respectively. and Let represent the second-order moment estimates of the gradients at sensor node i and time node l, respectively. This represents the regularization parameter used to prevent the model from overfitting. and This represents the weights used to control the first and second moments of the gradient; S430, Adaptive adjustment of the learning rate parameter, as detailed below. in, For learning rate, For adaptive learning rate parameters, This represents the number of iterations. These are the first-order moment weights and second-order moment weights after t iterations; S440, Update the latent vector and : Here, ε is a small constant used to prevent division by zero errors; It is the updated sensor node vector; It is the updated time node vector; S450, The predicted value of the data matrix that needs compensation is: in, The predicted vector value represents the measurement output: According to (1-2)-(1-13), the filter is defined in... The sensor measurement data received at any given time is: In formula (1-14), with Data This represents the sensor measurement data received by the filter, corresponding to the node. exist Instantaneous measurement value.

5. The recursive filtering method based on matrix decomposition for sparse sensor measurements according to claim 3, characterized in that: Step S600 specifically includes, S610, One-step prediction error covariance matrix and the filter error covariance matrix The following two recurrence equations are satisfied: in, This represents the external coupling configuration matrix for the i-th and z-th nodes of the sensor; as well as, in, ; exist Expand using Taylor's formula ,get: In the formula , These are the higher-order components in the Jacobian matrix and Taylor series expression, respectively; Will Represented as: in, To meet The time-varying matrix of unknown values, and Represent two known matrices; The filtering error is obtained from formulas (1-1), (1-14), and (1-15): S620. Consider the one-step prediction error covariance matrix in formulas (1-16) and (1-17) respectively. and the filter error covariance matrix ;make and Let be a positive scalar; suppose the following two recursive matrix equations are: in, ; It is the upper bound matrix of the filtering error covariance matrix of the i-th sensor node at time l; as well as, It has a positive definite solution. and initial conditions Then, the matrix yes The upper bound of the matrix yes The upper bound, namely: as well as, The filter gain matrix is ​​minimized using the upper bound matrix. The trace calculation, that is, in, .

6. The recursive filtering method based on matrix decomposition for sparse sensor measurements according to claim 2, characterized in that: When the recursive filtering method for the multi-robot localization system is applicable to a discrete nonlinear complex network model with various Gaussian noises, it includes the following steps: S100, constructing a network model, in, and These represent the pre-communication transmission steps. The state and measurement output at each moment; x i,l+1 This represents the state input of the i-th sensor at time l+1; Represents the external coupling configuration matrix; The inner coupling strength matrix is ​​represented as ; It is a zero-mean Gaussian random multiplicative noise variable with unit variance; and These are process noise and measurement noise, respectively. and Let be the zero-mean Gaussian white noise at node i at time l, and let their covariance matrices be respectively. and Assuming , and In any Time is irrelevant; It is a known nonlinear function; and All are known matrices; S200 employs low duty cycle scheduling to manage data transmission, dividing the sensor into... Each node, the measurement output before transmission is described as follows: In the formula Indicates in From time to time Measurement output obtained by sensor nodes; under low duty cycle scheduling Not in the sensor exist It is sent while the device is in a dormant state. S300, Order In the formula For the first The sensor node at the ... The measurement output of time; The update rules are as follows: S400, where, Represents random variables Describe whether a DoS attack has occurred, considering each node... The probability of being attacked is equal; for And it follows a Bernoulli distribution, with the probability distribution of its random variable being: in, It is a known constant, and , in, For calculation based on low duty cycle scheduling The periodicity of the sensor node, function It means l divided by Remainder; Duty Cycle for: In the formula, For sensors Activation time within a cycle; S500, the measurement output received by the filter, constructed according to formulas (2-2)-(2-5), is defined as follows: , ; S600, construct the following recursive filter: in, and Representing time respectively One step of prediction and status The filtered value; Is The filter gain is designed for the specific moment. S700 performs filter processing on complex nonlinear networks with multiplicative noise under low duty cycle scheduling when subjected to DoS attacks.

7. The recursive filtering method based on matrix decomposition for sparse sensor measurements according to claim 6, characterized in that: Step S700 specifically includes, S710, One-step prediction error covariance matrix and the filter error covariance matrix The following two recurrence equations are satisfied: in, Represents the external coupling coefficients of the i-th node and the z-th node of the sensor; and, in, ; exist Expand using Taylor's formula ,get: In the formula, , These are the higher-order components in the Jacobian matrix and Taylor series expression, respectively; Represented as: in, To meet The unknown time-varying matrix, and Represent two known matrices; The filtering error can be obtained from (2-1) and (2-6): S720. Consider the one-step prediction error covariance matrix in formulas (2-7) and (2-8) respectively. and the filter error covariance matrix ;make and Let be a positive scalar; suppose the following two recursive matrix equations are: in, ; It is the upper bound matrix of the filter error covariance matrix of the i-th sensor node at time l; as well as, It has a positive definite solution. and initial conditions Then, the matrix yes The upper bound of the matrix yes The upper bound, that is, as well as, The filter gain matrix is ​​minimized using the upper bound matrix. The trace calculation, that is, in, .

8. A recursive filtering system based on matrix decomposition for sparse sensor measurements, characterized in that: The system has a program module corresponding to the steps of any one of the claims 1-7 above, and executes the steps in the above-described recursive filtering method based on matrix decomposition in the case of sparse sensor measurements when it is run.

9. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program configured to, when invoked by a processor, implement the steps of the recursive filtering method based on matrix decomposition in the case of sparse sensor measurements as described in any one of claims 1-7.

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