Performance analysis method for error-tolerant data state estimator

Abstract

The invention provides a performance analysis method for an error-tolerant data state estimator, and relates to the technical field of information physical system remote estimation and communication. According to the method, parameters of a remote estimation system are preprocessed, evolution of estimation error covariance and a related parameter matrix are initialized, then the rank of the parameter matrix is used as a judgment condition of binary search, and the numerical value and precision of a stable boundary which is finally solved can be given after multiple iterations of the algorithm. According to the method, the boundary of the data accuracy to the stability is accurately solved according to the parameters of the system model, and the lower definite bound of the data accuracy for ensuring the remote estimation stability under the uncertain network factors in the information physical system is provided, so that the method has higher universality and practicability; according to the method, the stability of the estimator can be effectively judged, and support is provided for remote estimation algorithm design under the existence of network instability factors.

Description

Technical Field

[0001] The present invention relates to the technical field of cyber-physical system remote estimation and communication technology, and particularly relates to a method for performance analysis of a state estimator for tolerable error data. Background Art

[0002] A cyber-physical system (CPS) realizes real-time monitoring and control of complex systems by closely integrating physical processes with computing and communication technologies. The core of CPS lies in its high degree of integration and collaboration, enabling each subsystem to seamlessly cooperate and jointly complete complex tasks. Due to its ability to improve system efficiency and response speed, it has a wide range of applications, including intelligent manufacturing, smart cities, healthcare, etc., greatly promoting the digital transformation and intelligent upgrading of various industries. However, due to the existence of unstable factors such as communication interference and cyber attacks, data cannot be transmitted normally, and the remote estimation results diverge. Therefore, studying the change of stability with data correctness rate, that is, the stability boundary, is of great significance for ensuring the stability of remote estimation.

[0003] Existing remote estimation stability methods mainly focus on first constructing a Lyapunov function related to remote estimation error, and then determining the stability of the remote estimator according to its positive definiteness. However, this method is only a sufficient condition for determining stability and cannot more precisely solve the critical state of whether the remote estimation is stable. Limited by the construction method of the Lyapunov function, the solved remote estimation algorithm may exclude some originally feasible or even better-performing algorithms. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for performance analysis of a state estimator for tolerable error data aiming at the above-mentioned deficiencies of the prior art, which can accurately solve the data correctness rate boundary for ensuring the stability of the estimator under different degrees of network uncertainty factors.

[0005] To solve the above technical problems, the technical solution adopted by the present invention is:

[0006] A method for performance analysis of a state estimator for tolerable error data, comprising the following steps:

[0007] S1: Construct a dynamic model of the overall system to describe the change of the system physical process and the process of sensor data being collected and transmitted to the remote estimation end;

[0008] S2: Starting from the initial moment k = 1, the remote estimation end uses the Kalman filter estimation algorithm to calculate the corresponding error covariance and fuse the measurement data of the sensor end;

[0009] S3: Initialize the transfer matrix L of the error covarianceγk , randomly initialize a value for the data anomaly rate μ that satisfies ;

[0010] S4: Based on S3, initialize the prior estimation error covariance of the system's remote estimation end;

[0011] S5: According to the quantities initialized in S4 and the error covariance iteration formula, determine the stability of the remote estimation error;

[0012] S6: At this time, there are only two results for the determination of the remote estimation error stability: stable and unstable. According to the two results, use the bisection method to gradually approach the critical data anomaly rate for remote estimation stability, and obtain the output result of the final data anomaly rate μ;

[0013] S7: Output the result β = 1 - μ and the calculation accuracy 1 / 2 L , where L represents the number of iterations of the algorithm loop. At this time, the value of β is the critical data correct rate.

[0014] Furthermore, in step 1, the process state, measurement value, and noise influence are expressed in the form of the system state space as follows:

[0015] x k+1 = Ax k + w k (1)

[0016] y k = Cx k + v k (2)

[0017] where, x k+1 is the system state vector, y k is the sensor measurement vector; w k and v k respectively represent the system noise and measurement noise, and both are independent zero-mean Gaussian white noises; A and C respectively represent the system state transition matrix and the sensor measurement matrix, satisfying (A, C) is observable, controllable.

[0018] Furthermore, the calculation steps of the Kalman filter in step 2:

[0019] S2.1: Time update:

[0020]

[0021] S2.2: Measurement update:

[0022]

[0023] where, and represent the prior estimate value and the posterior estimate value of the system state respectively, and P k represent the corresponding prior error covariance and posterior error covariance respectively; z k is called the innovation, also known as the residual, which represents the difference between the current measurement value and the estimate of the current measurement value by the prior estimate at the previous moment; K k is the Kalman filter gain of the estimator, (·) T represents the transpose of the matrix; when the estimator tends to be stable, that is, when the number of iterations is large enough, the prior estimate error covariance satisfies The Kalman gain satisfies K = K k ; Q and R are the covariance matrices of the system noise w k and the measurement noise v k respectively;

[0024] After the number of iterations is large enough, the actual innovation received at the remote estimation end is adjusted to:

[0025]

[0026] where, γ k = 1 corresponds to the case where abnormal data is received at the remote estimation end, γ k = 0 corresponds to the data received under normal conditions; γ k follows a Bernoulli distribution and satisfies Pr{γ k = 1} = μ, where Pr{·} represents probability; b k is a Gaussian white noise with a mean of 0 and a variance of ∑ b ; T is the connection matrix between the true data and the abnormal data.

[0027] Furthermore, in the step 3, the initialization matrix formula is:

[0028]

[0029] The calculation methods of and μ are as follows:

[0030]

[0031] where, L 0 and L 1 are the transfer matrices of the estimator error covariance matrix when the data is correct and when the data is abnormal respectively, which are calculated separately according to the cases where γ k takes the value of 0 or 1 in the formula (10); represents the largest singular value of the matrix; ρ(·) represents the spectral radius of the matrix.

[0032] Furthermore, in the step 4, the prior estimate error covariance at the remote estimation end is Its iterative calculation formula is as follows:

[0033]

[0034] Among them, the linear operator defined on the positive definite matrix M is:

[0035]

[0036] All variable definitions are calculated according to the following formula:

[0037]

[0038]

[0039] Among them, Q 0 and Q 1 are the covariance matrices of the noise in the estimation error when receiving normal data and abnormal data respectively, which are calculated from the cases where γ takes values of 0 and 1 in k

[0040] Furthermore, the specific method of step 5 is as follows:

[0041] S5.1: Initialize variables:

[0042]

[0043] Among them, vec(·) is to vectorize the matrix column-wise; is an intermediate variable defined for determining the iterative stability of formula (11) in the next step, that is, an intermediate variable defined for determining the stability of the estimator;

[0044] S5.2: Initialize the following remote estimator stability determination rules:

[0045] Case 1: It is necessary to further determine the error covariance matrix at steady state through If is a positive definite matrix, then the estimator is stable, otherwise the estimator is unstable; where n is the dimension of the system state, that is, the dimension of x x k

[0046] Case 2: Then the estimator is unstable;

[0047] Case 3: Then the estimator is unstable.

[0048] Furthermore, the specific processing process in step 6 is as follows: ​​​

[0049] S6.1: Update μ and the value of μ;

[0050] When it is stable, it indicates that the current value of μ is too small, and update it according to the following rules:

[0051]

[0052] When it is unstable, it indicates that the current value of μ is too large, and update it according to the following rules:

[0053] μ ←μ (20)

[0054] μ←( μ +μ) / 2 (21)

[0055] S6.2: When the number of iterations is less than L, return to S3 for recalculation; otherwise, output the result μ and end the calculation.

[0056] The beneficial effects of adopting the above technical solutions are as follows: The performance analysis method for the estimator of tolerable error data provided by the present invention accurately solves the boundary of the data correct rate for stability according to the parameters of the system model, provides the infimum of the data correct rate to ensure the stability of remote estimation under uncertain network factors in the cyber-physical system, making the present invention more universal and practical; the method of the present invention can effectively determine the stability of the estimator and provides support for the design of remote estimation algorithms under network instability factors. Description of the Drawings

[0057] Figure 1 It is the structural block diagram of the remote estimation system provided by the embodiment of the present invention;

[0058] Figure 2 It is the flowchart of the performance analysis method for the estimator of tolerable error data provided by the embodiment of the present invention;

[0059] Figure 3 It is the schematic diagram of the data abnormality rate and the actual estimation error in the performance analysis method for the estimator of tolerable error data provided by the embodiment of the present invention. Detailed Embodiment

[0060] The following combines the drawings and embodiments to further describe in detail the specific embodiments of the present invention. The following embodiments are used to illustrate the present invention, but are not used to limit the scope of the present invention.

[0061] This embodiment is applicable to a linear discrete time-invariant system. Using the performance analysis method for the estimator of tolerable error data proposed by the present invention, the block diagram of the actual remote estimation system under investigation is as Figure 1As shown, the sensor sends the innovation of the measurement calculation for the physical process to the remote estimator through a perturbed channel. This method first preprocesses the parameters of the remote estimation system and initializes the evolution of the estimation error covariance and the relevant parameter matrices. Then, taking the rank of the parameter matrix as the judgment condition for binary search, the numerical value and accuracy of the stable boundary of the final solution can be given after multiple iterations of the algorithm.

[0062] The specific implementation steps are as Figure 2 shown. The method of this embodiment is described as follows.

[0063] S1: Construct a dynamic model of the overall system to describe the change of the system physical process and the process of sensor data collection and transmission to the remote estimation end. Represent the process state, measurement value, and noise influence in the form of the system state space as follows:

[0064] x k+1 = Ax k + w k (1)

[0065] y k = Cx k + v k 2

[0066] Wherein, x k+1 is the system state vector, and y k is the sensor measurement vector; w k and v k represent the system noise and measurement noise respectively, and the two are independent zero-mean Gaussian white noises. A and C represent the system state transition matrix and the sensor measurement matrix respectively, and satisfy (A, C) is observable, controllable.

[0067] S2: Starting from the initial time k = 1, the remote estimation end uses the Kalman filter estimation algorithm to calculate the corresponding error covariance and fuse the measurement data of the sensor end. The following are the calculation steps of the Kalman filter:

[0068] S2.1: Time update:

[0069]

[0070] S2.2: Measurement update:

[0071]

[0072] Wherein, and represent the prior estimate value and posterior estimate value of the system state respectively, and P krespectively represent the corresponding prior and posterior error covariances; z k is called the innovation, also known as the residual, which represents the difference between the current measurement value and the estimation of the current measurement value by the prior estimate at the previous moment. K k is the Kalman filter gain of the estimator, (·) T represents the transpose of the matrix; when the estimator tends to be stable, that is, when the number of iterations is large enough, the prior estimation error covariance satisfies The Kalman gain satisfies K = K k ; Q and R are the system noise w k and the measurement noise v k of the covariance matrices, respectively.

[0073] After the number of iterations is large enough, the actual innovation received at the remote estimation end is adjusted to:[[]]

[0074]

[0075] where γ k = 1 corresponds to the case where the remote estimation end receives abnormal data, and γ k = 0 corresponds to the data received under normal conditions. γ k follows a Bernoulli distribution and satisfies Pr{γ k = 1} = μ, where Pr{·} represents probability. b k is a Gaussian white noise with a mean of 0 and a variance of ∑ b ; T is the connection matrix between the true data and the abnormal data.

[0076] S3: Initialize the transition matrix of the error covariance:

[0077] Randomly initialize a value for the data anomaly rate μ that satisfies where and μ The calculation methods are as follows:

[0078]

[0079] where L 0 and L 1 are the transition matrices of the estimator error covariance matrix when the data is correct and when the data is abnormal, respectively, which are calculated separately for the cases where γ k takes the values 0 or 1 in formula (10); represents the largest singular value of the matrix; ρ(·) represents the spectral radius of the matrix.

[0080] S4: Based on S3, the prior estimation error covariance at the remote estimation end is Its iterative calculation formula:

[0081]

[0082] Among them, the linear operator defined on the positive definite matrix M is:

[0083]

[0084] All variables are defined and calculated according to the following formula:

[0085]

[0086]

[0087] Among them, Q 0 and Q 1 are the covariance matrices of the noise in the estimation error when receiving normal data and abnormal data respectively, and are calculated from the cases where γ takes values of 0 and 1 in k

[0088] S5: Determine the stability of the remote estimation error according to the quantities initialized in S4 and the error covariance iteration equation. The specific process is as follows:

[0089] S5.1: Initialize variables:

[0090]

[0091] Among them, vec(·) column-vectorizes the matrix; is an intermediate variable defined for determining the iteration stability of formula (11) in the next step, that is, an intermediate variable defined for determining the stability of the estimator.

[0092] S5.2: Initialize the following remote estimator stability determination rules:

[0093] Case 1: It is necessary to further determine the error covariance matrix at the steady state through If is a positive definite matrix, then the estimator is stable, otherwise the estimator is unstable; where n x is the dimension of the system state, that is, the dimension of x k ;

[0094] Case 2: Then the estimator is unstable;

[0095] Case 3: Then the estimator is unstable.

[0096] ​S6: At this time, there are only two results for the determination of the stability of the remote estimation error, namely stable and unstable. According to these two results, the bisection method is used to gradually approach the critical data anomaly rate for remote estimation stability. The specific processing process is as follows:

[0097] S6.1: For μ, Update the value of μ;

[0098] When it is stable, it indicates that the current value of μ is too small, and it is updated according to the following rules:

[0099]

[0100] When it is unstable, it indicates that the current value of μ is too large, and it is updated according to the following rules:

[0101] μ ←μ (20)

[0102] μ←( μ +μ) / 2 (21)

[0103] S6.2: When the number of iterations is less than L, return to S3 for recalculation; otherwise, output the result μ and end the calculation.

[0104] S7: Output the result β = 1 - μ and the calculation accuracy 1 / 2 L , and at this time, the value of β is the critical data correct rate.

[0105] As Figure 3 shown, after the estimator runs for a long time, the estimation error increases as the data anomaly rate increases and diverges when approaching a certain threshold. The present invention gives the upper and lower bounds of this threshold and uses an algorithm to solve the accurate value of this threshold, which is completely consistent with the simulation results.

[0106] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope defined by the claims of the present invention.

Claims

1. A performance analysis method for a tolerable error data state estimator, characterized in that: The following steps are involved: S1: Build a dynamic model of the overall system to describe the changes in the system's physical processes and the process of transmitting sensor data to the remote estimation end after acquisition; S2: Starting from the initial time k=1, the remote estimation end uses the Kalman filter estimation algorithm to calculate the corresponding error covariance and fuse the measurement data of the sensor end; S3: Initialize the error covariance transfer matrix L γk , randomly initialize the data anomaly rate μ to satisfy The value of S4: Based on S3, initialize the prior estimation error covariance of the remote estimation end of the system; S5: Determine the stability of the remote estimation error based on the quantity initialized in S4 and the error covariance iteration formula; S6: At this time, there are only two results for the remote estimation error stability judgment: stable and unstable. According to the two results, a dichotomy method is used to gradually approach the critical data anomaly rate of the remote estimation stability, and the output result of the final data anomaly rate μ is obtained; S7: Output result β = 1-μ and calculation accuracy 1 / 2 L , where L represents the number of iterations of the algorithm cycle, and the value of β is the critical data accuracy.

2. A performance analysis method for a tolerable error data state estimator according to claim 1, characterized in that: In step 1, the process state, measurement value and noise influence are expressed in the form of system state space as follows: x k+1 =Ax k +w k (1) y k =Cx k +v k (2) Among them, x k+1 is the system state vector, y k is the sensor measurement vector; w k and v k Represent the system noise and measurement noise respectively, both of which are independent zero-mean Gaussian white noise; A and C represent the system state transfer matrix and sensor measurement matrix respectively, satisfying (A, C) observability. Can be controlled.

3. The performance analysis method for a tolerable error data state estimator according to claim 2, characterized in that: The calculation steps of Kalman filtering in step 2 are: S2.1: Time update: S2.2: Measurement update: in, and denote the prior and a posteriori estimates of the system state, respectively. and P k They represent the corresponding prior error covariance and posterior error covariance respectively; z k It is called the new information, also called the residual, which represents the difference between the current measurement value and the prior estimate of the current measurement value at the previous moment; K k is the estimator Kalman filter gain, (·) T represents the transpose of the matrix; when the estimator tends to be stable, that is, the number of iterations is large enough, the prior estimation error covariance satisfies Kalman gain satisfies K = K k ; Q and R are the system noise w k and measurement noise v k The covariance matrix of After a sufficient number of iterations, the actual new information received at the remote estimation end is adjusted to: Among them, γ k = 1 corresponds to the case where the remote estimation end receives abnormal data, γ k =0 corresponds to the data received in normal state; γ k Obeying Bernoulli distribution, satisfying Pr{γ k =1}=μ, where Pr{·} represents probability; b k It has a mean of 0 and a variance of Σ b Gaussian white noise; T is the connection matrix between real data and abnormal data.

4. The performance analysis method for a tolerable error data state estimator according to claim 3, characterized in that: In step 3, the initialization matrix formula is: The calculation methods of and μ are as follows: Among them, L0 and L1 are the transfer matrices of the estimator error covariance matrix when the data is correct and the data is abnormal, respectively. k The cases where the value is 0 or 1 are calculated separately; represents the largest singular value of the matrix; ρ(·) represents the spectral radius of the matrix.

5. The performance analysis method for a tolerable error data state estimator according to claim 4, characterized in that: In step 4, the prior estimation error covariance of the remote estimation end is The iterative calculation formula is: Among them, the linear operator defined on the positive definite matrix M is: All variable definitions are calculated according to the following formula: Among them, Q0 and Q1 are the covariance matrices of the noise in the estimation error when receiving normal data and abnormal data, respectively, given by Medium k The values ​​are 0 and 1.

6. A performance analysis method for a tolerable error data state estimator according to claim 5, characterized in that: The specific method of step 5 is: S5.1: Initialize variables: Among them, vec(·) is to vectorize the matrix columns; It is the intermediate variable defined in the next step to determine the iterative stability of formula (11), that is, the intermediate variable defined to determine the stability of the estimator; S5.2: Initialize the following long-range estimator stability decision rule: Case 1: Need to further pass Determine the error covariance matrix in the steady state if is a positive definite matrix, then the estimator is stable, otherwise the estimator is unstable; where n x is the dimension of the system state, i.e. x k Dimensions; Case 2: Then the estimator is unstable; Case 3: Then the estimator is unstable.

7. A performance analysis method for a tolerable error data state estimator according to claim 6, characterized in that: The specific processing process in step 6 is as follows: S6.1: For μ, Update the value of μ; When it is stable, it means that the current value of μ is too small and is updated according to the following rules: If it is unstable, it means that the current value of μ is too large and is updated according to the following rules: μ ←μ (20) μ←( μ +μ) / 2 (21) S6.2: When the number of iterations is less than L, return to S3 and calculate again; otherwise, output the result μ and end the calculation.

Images