Distributed information physical system security state estimation method based on recursive least square and fading Kalman filtering

Abstract

The invention belongs to the field of information physical system security state estimation and fault diagnosis, and discloses a distributed information physical system security state estimation method based on recursive least square and fading Kalman filtering. Comprising the steps of establishing a distributed linear information physical system discrete model of which an actuator is attacked, establishing a subsystem attack signal recursive least square iteration model, establishing a subsystem fading Kalman filtering mathematical model, designing fading factors, designing subsystem weighting coefficients and performing data weighted fusion. The method provided by the invention can perform security state estimation on the system state vector and the attack signal in time and in real time, and has the advantages of strong adaptability, high accuracy, good stability and the like.

Description

Technical Field

[0001] The present invention belongs to the field of cyber-physical system (CPS) security state estimation and fault diagnosis, and relates to a false data injection (FDI) attack security state estimation method for a distributed linear CPS actuator based on recursive least squares and fading Kalman filtering. Background Art

[0002] Nowadays, cyber-physical systems are widely used in smart grids, automated transportation, medical equipment, industrial control and other fields. With the advancement of technology, the security of cyber-physical systems has become increasingly important because they rely not only on the reliability of physical devices, but also on the security of networks and software. In such systems, security state estimation is a key element, which aims to monitor and evaluate the system state in real time in order to detect potential security threats and failures in a timely manner. This process involves dynamic estimation of the system state, including processing of sensor data, model prediction and anomaly detection. Faced with increasingly complex attack methods and environmental changes, traditional security mechanisms are no longer sufficient to cope with all security challenges. Therefore, integrating advanced state estimation technology and data fusion methods will improve the security and stability of cyber-physical systems and ensure that they can maintain normal operation and make corresponding decisions in the face of malicious attacks, equipment failures or environmental changes. In this context, studying the security state estimation method of cyber-physical systems is of great significance to ensure the security, reliability and resilience of critical infrastructure.

[0003] False data injection attacks are a threat to cyber-physical systems and other network systems. Attackers inject forged or erroneous data into the system in order to disrupt the normal operation of the system, mislead decisions, or cause potential physical damage. False data injection attacks can be divided into attacks against sensors and attacks against actuators. Sensor attacks provide incorrect information to the control system by tampering with or forging sensor data. For example, attackers may send false temperature, pressure, or position data, causing the system to make incorrect decisions and control instructions. Such attacks are usually used to hide the true state or manipulate system behavior; actuator attacks directly target the system's execution mechanism. Attackers may interfere with or manipulate the operation of the actuator to make it perform incorrect behaviors. For example, attackers can affect the normal operation of the system by tampering with the signal, causing the motor to overload or close the valve. This type of attack intends to directly affect the physical process and may cause equipment damage or safety accidents. Therefore, developing effective detection and defense mechanisms, as well as real-time monitoring of system status and data authenticity, is an important research direction for protecting the security of cyber-physical systems. In the face of the increasingly complex background of modern cyber attacks, it is particularly important to strengthen the understanding and prevention of false data injection attacks.

[0004] In the prior art, the Chinese patent application publication number: CN112788049B discloses a method for detecting attacks on power cyber-physical systems based on adaptive Kalman filter. However, this method still has the following problems:

[0005] (1) A detection method for actuator attacks is given, but no effective method for estimating the security state of the system when it is attacked is given, including the estimation of the attack signal.

[0006] (2) The research object is a single simple cyber-physical system, and no corresponding security state estimation and data fusion methods for the more widely used distributed cyber-physical systems are given.

[0007] (3) The specific method for calculating the fading factor is not given, and the algorithm needs to be improved. Summary of the invention

[0008] The purpose of the present invention is to propose a scientific, reasonable, highly applicable and effective security state estimation method and data fusion method for false data injection attacks in view of the deficiencies in the prior art. The method is based on the recursive least squares method and the fading Kalman filter. When a distributed information-physical system is attacked, the security state of the system can be estimated in real time to ensure the security and reliability of the system and enhance the robustness of the system.

[0009] The technical solution of the present invention:

[0010] A distributed cyber-physical system safety state estimation method based on recursive least squares and fading Kalman filtering, the steps are as follows:

[0011] 1) Establish a discrete model of the i-th subsystem of a distributed linear CPS whose actuator is attacked by FDI, where i∈(1,...L); the discrete model is as follows:

[0012]

[0013] Among them, x i (k+1) is the state vector of subsystem i at time k+1, A i The (k) matrix is ​​the state transfer matrix of subsystem i at time k, which represents the state transfer relationship; i (k) is the state vector of subsystem i at time k; B i (k) is the input matrix of subsystem i at time k; u i (k) is the input signal of subsystem i at time k; a i (k) is the attack signal suffered by subsystem i at time k, ω i (k) is the process noise that conforms to Gaussian distribution; y i(k) is the measured value of the output of subsystem i at time k; C i (k) is the output matrix of subsystem i at time k; θ i (k) is the observation noise that conforms to Gaussian distribution; the above discrete model is general and representative;

[0014] 2) Establish a recursive least squares iterative model of the attack signal of the i-th subsystem

[0015] According to the above discrete model, the attack signal a is analyzed by recursive least squares method. i (k) Perform iterative estimation. The estimation of attack signals is an important part of security state estimation. The recursive least squares method can be used to update parameter estimates instantly when new data arrives, without recalculating all data, to handle the ever-changing attack signal data stream, and adjust parameters through the incremental update mechanism to achieve real-time estimation and accurate tracking of attack signals. The following is the specific formula:

[0016] γ i (k) = [I n -K i (k)C i (k)]A i (k)γ i (k-1)+[I n -K i (k)C i (k)]

[0017] Among them, γ i (k) is the attack error compensation matrix at time k, γ i (0) = 0; I n is the n-order unit matrix; K i (k) is the Kalman gain coefficient at time k; γ i (k-1) is the attack error compensation matrix at time k-1;

[0018] Ω i (k) = C i (k)A i (k)γ i (k-1)+C i (k)

[0019] Among them, Ω i (k) is an intermediate matrix that quantifies the reliability and confidence of the system state estimate;

[0020] Λ i (k) = [μ i ∑ i (k)+Ω i (k)S i(k-1)Ω i T (k)] -1

[0021] Among them, Λ i (k) is the transition matrix at time k, which is used to calculate the RLS gain coefficient Γ i (k); μ i is the forgetting factor, μ i ∈(0,1],μ i The smaller it is, the smaller the weight of historical data is;∑ i (k) represents the description matrix of the state uncertainty associated with the measurement at time k; S i (k-1) represents the error covariance matrix of the attack signal at time k-1; Ω i T (k) represents Ω i The transposed matrix of (k);

[0022] Γ i (k) = S i (k-1)Ω i T (k)Λ i (k)

[0023] Among them, Γ i (k) is the RLS gain coefficient, which is used to weigh the impact of new observation data in state estimation;

[0024]

[0025] in, is the estimated value of the attack signal at time k, realizing iteration; represents the estimated value of the attack signal at time k-1; ε i (k) is the new information sequence at time k;

[0026] ε i (k) = y i (k)-C i (k)[A i (k-1)x i (k-1)+B i (k-1)(u i (k-1)+a i (k-1))]

[0027]

[0028] Among them, S i (k) represents the error covariance matrix of the attack signal at time k; S i (0) = I is the identity matrix;

[0029] 3) Establish the mathematical model of the fading Kalman filter of the i-th subsystem

[0030] After completing the estimation of the attack signal, the fading Kalman filter is used to estimate the system state vector in real time. The fading Kalman filter adjusts the state covariance matrix by introducing a fading factor, so that the sensitivity to new data is enhanced and the sensitivity to old data is gradually reduced. The specific design method of the fading factor is described in the next step. Through the two processes of prediction and update, combined with the system model and observation data, it can effectively deal with the estimation error caused by inaccurate parameters and model uncertainty, quickly optimize parameters, and improve estimation accuracy. On this basis, the attack signal error compensation is introduced to obtain The specific process is as follows:

[0031]

[0032] in, It is the one-step prediction state estimate from time k-1 to time k; is the estimated value of the state at time k-1;

[0033]

[0034] Among them, P i p (k) is the one-step prediction state covariance matrix from time k-1 to time k; λ i (k) is the fading factor at time k; P i (k-1) is the state covariance matrix at time k-1; Q i (k-1) is the process noise covariance matrix at time k-1; Indicates A i Transpose of the (k-1) matrix;

[0035] K i (k) = P i p (k)C T i (k)[C i (k)P i p (k)C T i (k)+R i (k)] -1

[0036] Among them, K i (k) is the Kalman gain coefficient at time k; R i (k) is the observation noise covariance matrix at time k; C T i (k) indicates C i(k) Transpose of the matrix;

[0037]

[0038] in, is the estimated state value of subsystem i at time k;

[0039] P i (k) = [IK i (k)C i (k)]P i p (k)

[0040] Among them, P i (k) is the state covariance matrix of subsystem i at time k; I is the identity matrix;

[0041] 4) The fading factor λ of the i-th subsystem i Determination of (k)

[0042] Step 3) Complete the establishment of the fading Kalman filter mathematical model, and then design the fading factor λ i (k), so that it can realize the function of adjusting the degree of dependence on historical information; the selection of the fading factor is closely related to the system dynamic characteristics, noise level, and time series characteristics. The windowing method has good flexibility and adaptability in obtaining the fading factor; the windowing method defines a time window of fixed length, and data processing is performed within this window. The data in the window is used to calculate the fading factor, so that the dependence on historical observation data will decrease over time; this method provides a flexible mechanism, which can adjust the window size according to actual conditions, thereby affecting the calculation of the fading factor to adapt to data streams with different dynamic characteristics; the following is the specific formula:

[0043]

[0044] Among them, N i (k) and M i (k) are respectively related to the covariance matrix of the system Gaussian noise and the system state covariance matrix, and serve as intermediate variables for obtaining the fading factor;

[0045]

[0046] Among them, F i (k) is the theoretical value of the innovation covariance at time k; is the estimated value of the new information covariance obtained by the windowing method at time k; N is the length of the sliding window;

[0047]

[0048] Among them, max is the maximum value function; m is yi (k) is the dimension of the matrix; tr{} represents the trace of the matrix;

[0049] 5) Subsystem data fusion to obtain global state estimation

[0050] After the above steps, the estimation of the subsystem attack signal and the system state estimation have been completed. Next, the subsystem data is weighted and fused to obtain the final security state estimation of the entire system, which is the final result we need. First, the weighted matrix is ​​designed according to the accuracy and richness of the data provided by each subsystem, giving greater weight to data with higher accuracy, so that the data transmitted by each subsystem can be integrated in the best way. Here, the accuracy of the data is evaluated through the state covariance matrix between subsystems. The specific formula is as follows:

[0051]

[0052] Among them, W x (k) is the nL×n weighting matrix of the system state variables at time k; W i x (k) is the state weight coefficient of subsystem i at time k; e x =[I n ,...,I n ] T ; Indicates e x The transposed matrix of x (k) is the nL×nL state covariance matrix between different subsystems at time k, which is given by The symmetric positive definite matrix formed by For local estimation and The state cross-covariance matrix of ;

[0053] Similarly, for attack signals have

[0054]

[0055] Among them, W a (k) is the sL×s weighting matrix of the attack signal at time k; W i a (k) is the attack signal weighting coefficient of subsystem i at time k; e a =[I s ,...,I s ] T ; Indicates e a The transposed matrix of a (k) is the covariance matrix of the attack signal sL×sL between different subsystems at time k, which is given by The symmetric positive definite matrix formed by Local attack signal and The cross-covariance matrix of .

[0056] Beneficial effects of the present invention: The present invention novelly proposes a safe state estimation method based on the recursive least squares method and the fading Kalman filter when the distributed linear information-physical system is attacked by false data injection, provides a data fusion method between subsystems, and realizes parameter iteration. The method has the following advantages: First, it can monitor the system state in real time, effectively identify and filter false data, thereby significantly improving the accuracy of state estimation. Through the recursive least squares method, the system can converge quickly in the continuously updated data stream to ensure the timeliness and reliability of state estimation. Secondly, the introduction of the fading Kalman filter enables the system to flexibly adjust the state estimation strategy in response to multiple attacks or emergencies, reduce dependence on historical data, and further enhance the adaptability of the system. In addition, the method realizes data fusion between subsystems, promotes information sharing and coordination, and improves the operating efficiency of the overall system. This collaborative working mechanism not only strengthens the system's ability to respond to abnormal situations, but also improves the robustness of the system and reduces the risks caused by single point failures. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 It is a flow chart of a method for estimating the safe state of a distributed information-physical system based on recursive least squares and fading Kalman filtering of the present invention. DETAILED DESCRIPTION

[0058] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0059] In a basic implementation of the present invention, it can be divided into five steps. The purpose of the present invention is to provide a method for estimating the security state of a cyber-physical system based on a recursive least squares method and a fading Kalman filter to cope with false data injection attacks and other potential security threats faced by actuators in distributed cyber-physical systems. The specific implementation steps are as follows:

[0060] Step 1: Based on the characteristics of the distributed linear cyber-physical system, a discrete time model is constructed. The model should fully consider the relationship between various sensors and actuators in the system and their interactions to ensure that the dynamic behavior of the system can be accurately described. In this process, it is necessary to define the state vector, observation vector, and control input of the system, and link these variables through appropriate state equations and observation equations to achieve a comprehensive description of the cyber-physical system.

[0061]

[0062] Here, we take the most common standard distributed linear information-physical system as an example to establish a state space discrete model of its subsystems. The distributed linear information-physical system has L subsystems, and the model is general and representative. i (k+1) is the state vector of subsystem i at time k+1, x i (k+1) dimension is n; A i The (k) matrix is ​​the state transfer matrix of subsystem i at time k, which represents the state transfer relationship; i (k) is the state vector of subsystem i at time k; B i (k) is the input matrix of subsystem i at time k; u i (k) is the input signal of subsystem i at time k; a i (k) represents the attack signal acting on the actuator of the i-th subsystem at time k, a i The dimension of (k) is s; y i (k) is the measured value of the output of subsystem i at time k, which is the data we directly measure through sensors; C i (k) is the output matrix of subsystem i at time k; other interferences and noises are uniformly expressed as process noise ω that conforms to Gaussian distribution i (k) and observation noise θ i (k). The task of the present invention is to obtain x based on this model i (k) and a i (k) The final safety state estimation is performed and the estimation algorithm is guaranteed to be accurate and real-time.

[0063] Step 2: When the actuator is subjected to an attack signal, model the attack signal. Identify the characteristics of the attack signal and its impact on system performance, and design a corresponding model to describe the changes in these attack signals. Apply the recursive least squares algorithm to iteratively update the attack signal step by step to obtain a more accurate estimate of the attack signal. This step involves the repeated application and real-time update of historical data, thereby continuously improving the efficiency of identifying attack signals.

[0064] Υ i (k) = [I n -K i (k)C i (k)]A i (k)Υ i (k-1)+[I n -K i (k)C i (k)]

[0065] Among them, i(k) is the attack error compensation matrix at time k, Υ i (0) = 0; I n is the n-order unit matrix; K i (k) is the Kalman gain coefficient at time k; i (k-1) is the attack error compensation matrix at time k-1;

[0066] Ω i (k) = C i (k)A i (k)γ i (k-1)+C i (k)

[0067] Among them, Ω i (k) is an intermediate matrix that quantifies the reliability and confidence of the system state estimate;

[0068] Λ i (k) = [μ i Σ i (k)+Ω i (k)S i (k-1)Ω i T (k)] -1

[0069] Among them, Λ i (k) is the transition matrix at time k, which is used to calculate the RLS gain coefficient Γ i (k); μ i is the forgetting factor, μ i ∈(0,1],μ i The smaller it is, the smaller the weight of historical data is;∑ i (k) represents the description matrix of the state uncertainty associated with the measurement at time k; S i (k-1) represents the error covariance matrix of the attack signal at time k-1; Ω i T (k) represents Ω i The transposed matrix of (k);

[0070] Γ i (k) = S i (k-1)Ω i T (k)Λ i (k)

[0071] Among them, Γ i (k) is the RLS gain coefficient, which is used to weigh the impact of new observation data in state estimation;

[0072]

[0073] in, is the estimated value of the attack signal at time k, realizing iteration; represents the estimated value of the attack signal at time k-1; ε i (k) is the new information sequence at time k;

[0074] ε i (k) = y i (k)-C i (k)[A i (k-1)x i (k-1)+B i (k-1)(u i (k-1)+a i (k-1))]

[0075]

[0076] Among them, S i (k) represents the error covariance matrix of the attack signal at time k; S i (0) = I is the identity matrix;

[0077] Here we get the estimated value of the attack signal of subsystem i The expression of the error covariance matrix of the attack signal is given. i (k) can evaluate the information gain brought by the new measurement data. If the new information sequence is large, it means that the new measurement information is significantly different from the prediction, which may provide important update information for the system; on the contrary, if the new information sequence is small, it means that the current estimate is very close to the actual measurement, and the new measurement contributes little to the state update.

[0078] Step 3: Use the fading Kalman filter to estimate the system state vector in real time. The fading Kalman filter can effectively suppress noise and improve the estimation accuracy by continuously updating the state estimate. In this process, it is necessary to set appropriate initial conditions and the covariance matrix of process noise and observation noise to ensure that the filter can converge to the correct state estimate in a dynamically changing environment. At the same time, a recursive update process is implemented so that each new observation value can optimize the current state estimate.

[0079]

[0080] in, It is the one-step prediction state estimate from time k-1 to time k; is the estimated value of the state at time k-1;

[0081]

[0082] Among them, P ip (k) is the one-step prediction state covariance matrix from time k-1 to time k; λ i (k) is the fading factor at time k; P i (k-1) is the state covariance matrix at time k-1; Q i (k-1) is the process noise covariance matrix at time k-1; Indicates A i Transpose of the (k-1) matrix;

[0083] K i (k) = P i p (k)C T i (k)[C i (k)P i p (k)C T i (k)+R i (k)] -1

[0084] Among them, K i (k) is the Kalman gain coefficient at time k; R i (k) is the observation noise covariance matrix at time k; C T i (k) indicates C i (k) Transpose of the matrix;

[0085]

[0086] in, is the estimated state value of subsystem i at time k;

[0087] P i (k) = [IK i (k)C i (k)]P i p (k)

[0088] Among them, P i (k) is the state covariance matrix of subsystem i at time k; I is the identity matrix;

[0089] Step 4: In the process of estimating the state vector, obtain the fading factor. The fading factor is required to adjust the memory capacity of the filter, so that its influence on older observation information is gradually reduced, while being more sensitive to new information, and ensuring the stability and convergence of the filter.

[0090]

[0091] Among them, Ni (k) and M i (k) are respectively related to the covariance matrix of the system Gaussian noise and the system state covariance matrix, and serve as intermediate variables for obtaining the fading factor;

[0092]

[0093] Among them, F i (k) is the theoretical value of the innovation covariance at time k; is the estimated value of the new information covariance obtained by the windowing method at time k; N is the length of the sliding window;

[0094]

[0095] Among them, max is the maximum value function; m is y i (k) is the dimension of the matrix; tr{} represents the trace of the matrix;

[0096] Step 5: Design the weighting coefficients of the subsystems and weight the subsystem data to get the final estimate. Different weighting coefficients are assigned to each subsystem according to its importance in the overall estimate and its signal quality to optimize the result of data fusion. After weighted processing, the data transmitted by each subsystem can be integrated in the best way, and finally a comprehensive state estimate is output. This process needs to consider the flexibility and scalability of the system architecture so that new subsystems can be seamlessly integrated when they are connected.

[0097]

[0098] Among them, W x (k) is the nL×n weighting matrix of the system state variables at time k; W i x (k) is the state weight coefficient of subsystem i at time k; e x =[I n ,...,I n ] T ; Indicates e x The transposed matrix of x (k) is the nL×nL state covariance matrix between different subsystems at time k, which is given by The symmetric positive definite matrix formed by For local estimation and The state cross-covariance matrix of ;

[0099] Similarly, for attack signals have

[0100]

[0101] Among them, W a (k) is the sL×s weighting matrix of the attack signal at time k; W i a (k) is the attack signal weighting coefficient of subsystem i at time k; e a =[I s ,...,I s ] T ; Indicates e a The transposed matrix of a (k) is the covariance matrix of the attack signal sL×sL between different subsystems at time k, which is given by The symmetric positive definite matrix formed by Local attack signal and The cross-covariance matrix of ;

[0102] At this point, the final safety state estimate of the distributed linear cyber-physical system is obtained. And the state estimation of the attack signal And the respective state cross-covariance matrices are given;

[0103] The above is based on the ideal embodiment of the present invention. Through the above description, those skilled in the art can make various changes and modifications without departing from the technical concept of the present invention. The technical scope of the present invention is not limited to the content of the specification, and its technical scope must be determined according to the scope of the claims.

Claims

1. A method for estimating the safety state of a distributed cyber-physical system based on recursive least squares and fading Kalman filtering, characterized in that: Here are the steps: 1) Establish a discrete model of the i-th subsystem of a distributed linear CPS whose actuator is attacked by FDI, where i∈(1,...L); the discrete model is as follows: Among them, x i (k+1) is the state vector of subsystem i at time k+1, A i The (k) matrix is ​​the state transfer matrix of subsystem i at time k, which represents the state transfer relationship; i (k) is the state vector of subsystem i at time k; B i (k) is the input matrix of subsystem i at time k; u i (k) is the input signal of subsystem i at time k; a i (k) is the attack signal suffered by subsystem i at time k, ω i (k) is the process noise that conforms to Gaussian distribution; y i (k) is the measured value of the output of subsystem i at time k; C i (k) is the output matrix of subsystem i at time k; is the observation noise that conforms to Gaussian distribution; 2) Establish a recursive least squares iterative model of the attack signal of the i-th subsystem According to the above discrete model, the attack signal a is analyzed by recursive least squares method. i (k) Perform iterative estimation. The estimation problem of attack signals is an important part of security state estimation. The recursive least squares method is used to update parameter estimates instantly when new data arrives, without recalculating all data to handle the ever-changing attack signal data stream, and the parameters are adjusted through the incremental update mechanism to achieve real-time estimation and accurate tracking of attack signals. The specific formula is as follows: γ i (k)=[I n -K i (k)C i (k)]A i (k)γ i (k-1)+[I n -K i (k)C i (k)] Among them, i (k) is the attack error compensation matrix at time k, Υ i (0) = 0; I n is the n-order unit matrix; K i (k) is the Kalman gain coefficient at time k; i (k-1) is the attack error compensation matrix at time k-1; Oh i (k)=C i (k)A i (k)Y i (k-1)+C i (k) Among them, Ω i (k) is an intermediate matrix that quantifies the reliability and confidence of the system state estimate; Among them, Λ i (k) is the transition matrix at time k, which is used to calculate the RLS gain coefficient Γ i (k); μ i is the forgetting factor, μ i ∈(0,1],μ i The smaller it is, the smaller the weight of historical data is;∑ i (k) represents the description matrix of the state uncertainty associated with the measurement at time k; S i (k-1) represents the error covariance matrix of the attack signal at time k-1; Ω i T (k) represents Ω i The transposed matrix of (k); C i (k)=S i (k-1)Ω i T (k)L i (k) Among them, Γ i (k) is the RLS gain coefficient, which is used to weigh the impact of new observation data in state estimation; in, is the estimated value of the attack signal at time k, realizing iteration; represents the estimated value of the attack signal at time k-1; ε i (k) is the new information sequence at time k; ε i (k)=y i (k)-C i (k)[A i (k-1)x i (k-1)+B i (k-1)(u i (k-1)+a i (k-1))] Among them, S i (k) represents the error covariance matrix of the attack signal at time k; S i (0) = I is the identity matrix; 3) Establish the mathematical model of the fading Kalman filter of the i-th subsystem After completing the estimation of the attack signal, the fading Kalman filter is used to estimate the system state vector in real time. The fading Kalman filter adjusts the state covariance matrix by introducing a fading factor, so that the sensitivity to new data is enhanced and the sensitivity to old data is gradually reduced. Through the two processes of prediction and update, combined with the system model and observation data, the estimation error caused by inaccurate parameters and model uncertainty can be effectively dealt with, the parameters can be quickly optimized, and the estimation accuracy can be improved. On this basis, the attack signal error compensation is introduced to obtain The specific process is as follows: in, is the one-step predicted state estimate from time k-1 to time k; is the estimated value of the state at time k-1; in, is the one-step prediction state covariance matrix from time k-1 to time k; i (k) is the fading factor at time k; P i (k-1) is the state covariance matrix at time k-1; Q i (k-1) is the process noise covariance matrix at time k-1; Indicates A i Transpose of the (k-1) matrix; Among them, K i (k) is the Kalman gain coefficient at time k; R i (k) is the observation noise covariance matrix at time k; C T i (k) indicates C i (k) Transpose of the matrix; in, is the estimated state value of subsystem i at time k; Among them, P i (k) is the state covariance matrix of subsystem i at time k; I is the identity matrix; 4) The fading factor λ of the i-th subsystem i Determination of (k) Design to obtain the fading factor λ i (k), so that it has the function of adjusting the degree of dependence on historical information; the selection of the fading factor is closely related to the system dynamic characteristics, noise level and time series characteristics. The windowing method has good flexibility and adaptability in obtaining the fading factor; the windowing method defines a time window of fixed length, and data processing is performed within this window. The data in the window is used to calculate the fading factor, so that the dependence on historical observation data will decrease over time; the specific formula is as follows: Among them, N i (k) and M i (k) are respectively related to the covariance matrix of the system Gaussian noise and the system state covariance matrix, and serve as intermediate variables for obtaining the fading factor; Among them, F i (k) is the theoretical value of the innovation covariance at time k; is the estimated value of the new information covariance obtained by the windowing method at time k; N is the length of the sliding window; Among them, max is the maximum value function; m is y i (k) is the dimension of the matrix; tr{} represents the trace of the matrix; 5) Subsystem data fusion to obtain global state estimation The weighted fusion of subsystem data can obtain the final safety state estimation of the entire system. First, the weighted matrix is ​​designed according to the accuracy and richness of the data provided by each subsystem, giving greater weight to data with higher accuracy, so that the data transmitted by each subsystem can be integrated in the best way, and the accuracy of the data is evaluated through the state covariance matrix between subsystems. The specific formula is as follows: Among them, W x (k) is the nL×n weighting matrix of the system state variables at time k; is the state weight coefficient of subsystem i at time k; e x =[I n ,...,I n ] T ; Indicates e x The transposed matrix of x (k) is the nL×nL state covariance matrix between different subsystems at time k, which is given by i,j=1,2,...,L constitutes a symmetric positive definite matrix; For local estimation and The state cross-covariance matrix of ; Similarly, for attack signals have Among them, W a (k) is the sL×s weighting matrix of the attack signal at time k; is the weighted coefficient of the attack signal of subsystem i at time k; e a =[I s ,...,I s ] T ; Indicates e a The transposed matrix of a (k) is the covariance matrix of the attack signal sL×sL between different subsystems at time k, which is given by i,j=1,2,...,L constitutes a symmetric positive definite matrix; Local attack signal and The cross-covariance matrix of .

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