Linear system delay probability estimation method based on improved Kalman filtering algorithm

Abstract

The invention discloses a linear system delay probability estimation method based on an improved Kalman filtering algorithm, and belongs to the technical field of signal processing. By introducing the discrete Bernoulli random variable and constructing the layered Gaussian state space model, complex probability distribution is converted into an easy-to-process form, efficient joint estimation can be conveniently realized, unknown and time-varying delay probability can be adaptively estimated on line, the precision and stability of state estimation can be improved, and the method is suitable for large-scale popularization and application. The method is suitable for a linear system with one-step random delay measurement, and is especially suitable for state estimation of a visual inertial odometer (VIO) system.

Description

Technical Field

[0001] The present invention relates to the technical field of signal processing, and in particular to a linear system delay probability estimation method based on an improved Kalman filtering algorithm. Background Art

[0002] Kalman filter (KF), as a real-time online recursive state estimator, has been widely used in engineering fields such as signal processing, target tracking, navigation control, etc. KF is essentially a minimum variance estimator for linear systems with Gaussian noise distribution. It can obtain the optimal estimate of the state through a series of recursive calculations in the presence of process noise and measurement noise. However, in practical applications, especially in systems that transmit signals or data through the network, signals or data often encounter random delays, which can seriously affect the estimation accuracy and stability of KF.

[0003] To deal with the problem of random delay, researchers have proposed a variety of improved Kalman filters. For example, for nonlinear systems with one-step or two-step random delayed measurements (RDM), extended Kalman filters (EKF) and unscented Kalman filters (UKF) are designed, in which independent Bernoulli random variables are used to model the RDM. These methods improve the robustness and adaptability of the filter to a certain extent, but most of them assume that the delay probability is known. However, in linear systems with one-step random delayed measurements, especially in state estimation of visual inertial odometry (VIO) systems, the delay probability of RDM is often unknown and time-varying, which greatly limits the performance of existing methods.

[0004] In summary, there is an urgent need for a Kalman filter that can online estimate the probability of unknown and time-varying delays to improve the accuracy of state estimation in a random delay environment. Summary of the invention

[0005] The purpose of the present invention is to overcome the deficiencies in the prior art and to provide a linear system delay probability estimation method based on an improved Kalman filter algorithm, which can adaptively estimate unknown and time-varying delay probabilities online and improve the accuracy and stability of state estimation.

[0006] To achieve the above object, the present invention is implemented by adopting the following technical solutions:

[0007] On the one hand, the present invention provides a linear system delay probability estimation method based on an improved Kalman filter algorithm, comprising:

[0008] Acquire state data of a linear system; use the state data of the linear system as a linear state space model;

[0009] Adding a discrete Bernoulli random variable to the measurement equation of the linear state space model, and using the probability of the discrete Bernoulli random variable as the delay probability of the linear system;

[0010] Converting the probability density function of the actual measurement value in the linear state space model into an exponential product form, and constructing a hierarchical Gaussian state space model according to the exponential product form; using the hierarchical Gaussian state space model, expanding the state vector in the linear state space model, and using the discrete Bernoulli random variable and the delay probability as newly added states of the hierarchical Gaussian state space model;

[0011] The variational Bayes method and the state extension method are used to predict and update the extended state vector, the delay probability and the posterior shape parameters of the delay probability, and the delay probability estimate of the linear system is obtained.

[0012] Optionally, the linear state space model is expressed as:

[0013] ;

[0014] In the formula, is the discrete time index; At discrete time points The state vector on ; At discrete time points The state transfer matrix on ; At discrete time points The state vector on ; At discrete time points The process noise vector on ; At discrete time points The ideal measurement vector on ; At discrete time points The measurement matrix on ; At discrete time points The measurement noise vector on ; At discrete time points The actual measurement vector on ; At discrete time points Discrete Bernoulli random variables on ; At discrete time points The ideal measurement vector on .

[0015] Optionally, the hierarchical Gaussian state space model is expressed as:

[0016] ;

[0017] ;

[0018] ;

[0019] In the formula, is a hierarchical Gaussian state space model; It is in the form of exponential product; is the weighted sum form of Gaussian distribution; At discrete time points The actual measurement vector on ; At discrete time points The extended state vector on ; At discrete time points Discrete Bernoulli random variables on ; is a Gaussian distribution; At discrete time points The measurement matrix on ; At discrete time points The state vector on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The measurement matrix on ; At discrete time points The state vector on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The probability of delay on ; is the probability density function of the measurement error when there is a delay in the measurement; is the probability density function of the measurement error when there is no delay in the measurement; is the probability mass function.

[0020] Optionally, a variational Bayes method and a state extension method are used to update the extended state vector, the delay probability, and the posterior shape parameter of the delay probability to obtain an estimated value of the delay probability of the linear system, including:

[0021] Using the forgetting factor to expand the prior shape parameters, calculate the prior probability density function of the delay probability and the one-step prediction mean and covariance matrix of the extended state vector; process the extended state vector, the discrete Bernoulli random variable and the delay probability to obtain the posterior probability density function of the delay probability;

[0022] The extended state vector and its error covariance matrix are corrected using the actual measured values ​​in the linear state space model and the posterior probability density function, and the delay probability and its posterior shape parameters are updated using the corrected extended state vector and its error covariance matrix to obtain the delay probability estimate of the linear system.

[0023] Optionally, the prior probability density function of the delay probability is expressed as:

[0024] ;

[0025] ;

[0026] In the formula, is the prior probability density function of the delay probability; is Beta distribution; At discrete time points The probability of delay on ; and At discrete time points Upper delay probability The shape parameter of the prior Beta distribution of ; For the forgetting factor; and At discrete time points Upper delay probability The shape parameter of the posterior Beta distribution of .

[0027] Optionally, the one-step prediction mean and covariance matrix of the extended state vector are expressed as:

[0028] ;

[0029] ;

[0030] In the formula, At discrete time points The one-step predicted mean vector of the upper expanded state vector; At discrete time points The covariance matrix of the expanded state vector; At discrete time points The one-step prediction mean of the upper expanded state vector; At discrete time points The estimated value of the state on ; At discrete time points The one-step prediction error covariance matrix of the expanded state vector; At discrete time points The expanded state vector and at discrete time points The cross-covariance matrix between the expanded state vectors; At discrete time points The one-step forecast error covariance matrix on ; At discrete time points The state transfer matrix on ; At discrete time points The process noise covariance matrix on .

[0031] Optionally, the posterior probability density function of the delay probability is expressed as:

[0032] ;

[0033] In the formula, is the number of iterations; is a Gaussian distribution; At discrete time points The actual measurement vector after expansion; At discrete time points The expanded measurement matrix above; At discrete time points The upper corrected measurement noise covariance matrix; At discrete time points The one-step predicted mean vector of the upper expanded state vector; At discrete time points The covariance matrix of the expanded state vector; At discrete time points The measurement matrix on ; At discrete time points The measurement matrix on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The measurement noise covariance matrix on ; is the expected value; At discrete time points The extended state vector after the upper correction; At discrete time points The error covariance matrix after the upper correction; is the estimated value of the filter state; is the one-step smoothing estimate; is the error covariance matrix corresponding to the filter state estimate; is the cross covariance matrix; is the error covariance matrix corresponding to the one-step smoothing estimate.

[0034] Optionally, the step of determining the modified extended state vector and its error covariance matrix includes:

[0035] set up , using the state expansion method, the posterior probability density function is expressed as:

[0036] ;

[0037] in,

[0038] ;

[0039] ;

[0040] ;

[0041] After updating to Gaussian distribution, the modified extended state vector and its error covariance matrix are expressed as:

[0042] ;

[0043] In the formula, is the number of iterations; is a Gaussian distribution; At discrete time points The actual measurement vector after expansion; At discrete time points The expanded measurement matrix above; At discrete time points The upper corrected measurement noise covariance matrix; At discrete time points The one-step forecast mean vector on ; At discrete time points The covariance matrix of ; At discrete time points The measurement matrix on ; At discrete time points The measurement matrix on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The measurement noise covariance matrix on ; is the expected value; At discrete time points The extended state vector after the upper correction; At discrete time points The error covariance matrix after the upper correction; is the estimated value of the filter state; is the one-step smoothing estimate; is the error covariance matrix corresponding to the filter state estimate; is the cross covariance matrix; is the error covariance matrix corresponding to the one-step smoothing estimate.

[0044] Optionally, the step of determining the delay probability estimate value of the linear system includes:

[0045] set up , then the posterior probability density function is a Bernoulli distribution, expressed as:

[0046] ;

[0047] in,

[0048] ;

[0049] expect It is expressed as:

[0050] ;

[0051] set up , then the posterior probability density function is Beta distribution, and the delay probability estimate of the linear system is expressed as:

[0052] ;

[0053] in,

[0054] ;

[0055] ;

[0056] In the formula, is the posterior probability density function when the Bernoulli random variable is 1; is the posterior probability density function when the Bernoulli random variable is 0; is the normalization constant; is an exponential function; is the trace operation of the matrix; At a given discrete time point Observational data on and the current state estimate The expected value of the covariance matrix of the differences between observed and predicted data under the condition of ; At a given discrete time point Observational data on and discrete time points Observational data on The expected value of the covariance matrix of the difference between the observations at two moments under the condition of ; and When the measurement has no delay ( =0), there is a delay in measurement ( =1) when the expected value of the measurement noise covariance matrix; is the number of matrix rows or columns; is the identity matrix; is Beta distribution; and are the shape parameters of the Beta distribution after the i+1th iteration; and At discrete time points When the delay probability The shape parameter of the prior Beta distribution of ; is the double gamma function.

[0057] On the other hand, the present invention provides a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, the steps of the linear system delay probability estimation method based on the improved Kalman filter algorithm described in the first aspect are implemented.

[0058] Beneficial Effects

[0059] Compared with the prior art, the present invention has the following beneficial effects:

[0060] The present invention converts complex probability distribution into an easily tractable form by introducing discrete Bernoulli random variables and constructing a hierarchical Gaussian state space model, which facilitates efficient joint estimation. The variational Bayesian method and the state expansion method can adaptively estimate the unknown and time-varying delay probability online, improve the accuracy and stability of state estimation, and are suitable for linear systems with one-step random delay measurement, especially for state estimation of visual inertial odometry (VIO) systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 FIG. 1 is a flow chart of a linear system delay probability estimation method based on an improved Kalman filter algorithm in an embodiment of the present invention. DETAILED DESCRIPTION

[0062] The technical solution of the present invention is described in detail below through the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations on the technical solution of the present invention. The embodiments of the present invention and the technical features in the embodiments may be combined with each other unless there is a conflict.

[0063] The term "and / or" is only a way to describe the association relationship of related objects, indicating that there can be three relationships. For example, A and / or B can mean: A exists alone, A and B exist at the same time, and B exists alone. In addition, the character " / " generally indicates that the related objects are in an "or" relationship.

[0064] Example 1

[0065] like Figure 1 As shown, this embodiment introduces a linear system delay probability estimation method based on an improved Kalman filter algorithm, which specifically includes the following steps:

[0066] Acquire state data of a linear system; use the state data of the linear system as a linear state space model, and set an initial state estimate, an initial error covariance matrix, and a priori shape parameters of delay probability.

[0067] A discrete Bernoulli random variable is added to the measurement equation of the linear state space model, and the probability of the discrete Bernoulli random variable is used as the delay probability of the linear system.

[0068] The probability density function of the actual measurement value in the linear state space model is converted into an exponential product form, and a hierarchical Gaussian state space model is constructed according to the exponential product form; the state vector in the linear state space model is expanded by using the hierarchical Gaussian state space model, and the discrete Bernoulli random variable and the delay probability are used as new states of the hierarchical Gaussian state space model.

[0069] The variational Bayes method and the state extension method are used to update the extended state vector, the delay probability and the posterior shape parameters of the delay probability, and the delay probability estimate of the linear system is obtained.

[0070] When this embodiment is applied, by introducing discrete Bernoulli random variables and constructing a hierarchical Gaussian state space model, complex probability distributions are converted into an easy-to-handle form, which facilitates efficient joint estimation. By using a variational Bayesian method and a state expansion method, it is possible to adaptively estimate unknown and time-varying delay probabilities online, thereby improving the accuracy and stability of state estimation. This embodiment is suitable for linear systems with one-step random delay measurements, and is particularly suitable for state estimation of visual inertial odometry (VIO) systems.

[0071] Example 2

[0072] Based on Example 1, this example also has the following design:

[0073] Acquire state data of the linear system; use the state data of the linear system as a linear state space model to clarify key variables and their relationships.

[0074] The linear state space model is expressed as:

[0075] ;

[0076] In the formula, is the discrete time index; At discrete time points The state vector on ; At discrete time points The state transfer matrix on ; At discrete time points The state vector on ; At discrete time points The process noise vector on ; At discrete time points The ideal measurement vector on ; At discrete time points The measurement matrix on ; At discrete time points The measurement noise vector on ; At discrete time points The actual measurement vector on ; At discrete time points Discrete Bernoulli random variables on ; At discrete time points The ideal measurement vector on . The covariance matrices are and , represents the Kronecker delta function.

[0077] Add a discrete Bernoulli random variable with a value of 0 or 1 to the measurement equation of the linear state space model ,Right now:

[0078] ;

[0079] in, represents the delay probability at discrete time point k. The probability of the discrete Bernoulli random variable is used as the delay probability of the linear system.

[0080] Using the above formula, the probability density function (PDF) of the actual measurement value in the linear state space model can be expressed as a weighted sum of two Gaussian distributions:

[0081] .

[0082] However, because the PDF of the actual measurement value in the form of the weighted sum of two Gaussian distributions is neither closed nor conjugate, it is not possible to directly use Bayesian inference to estimate the state vector and unknown parameters. Therefore, its weighted sum form is converted into an exponential product form, and then a hierarchical Gaussian state space model (SSM) is established. Using the probability mass function (PMF):

[0083] .

[0084] The hierarchical Gaussian state space model rewrites the PDF as:

[0085] .

[0086] It can then be converted into a stratified Gaussian form as follows:

[0087] .

[0088] in, Represents a discrete time point The extended state vector on ; is a Gaussian distribution; At discrete time points The measurement matrix on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The measurement noise covariance matrix on ; is the probability density function of the measurement error when there is a delay in the measurement; is the probability density function of the measurement error when there is no delay in the measurement.

[0089] The hierarchical Gaussian state space model is used to expand the state vector in the linear state space model, and the discrete Bernoulli random variable and the delay probability are used as newly added states of the hierarchical Gaussian state space model.

[0090] The variational Bayes method and the state extension method are used to predict and update the extended state vector, the delay probability and the posterior shape parameters of the delay probability, and the delay probability estimate of the linear system is obtained.

[0091] Considering that Beta distribution is the conjugate prior distribution of Bernoulli distribution, in order to ensure conjugation, Beta distribution is selected as the prior probability density function (PDF) of delay probability, that is:

[0092] ;

[0093] in, is Beta distribution; and represents the prior shape parameter, that is and At discrete time points Upper delay probability The shape parameters of the prior Beta distribution are obtained by expanding the posterior shape parameters by a factor ρ. The specific expressions are as follows:

[0094]

[0095] in, and At discrete time points The posterior shape parameter on and At discrete time points Upper delay probability is the shape parameter of the posterior Beta distribution, while ρ∈(0,1] describes the degree of time fluctuation and is a forgetting factor.

[0096] Extended State Vector The one-step prediction probability density function (PDF) of is modeled as a Gaussian distribution, namely:

[0097] ;

[0098] At discrete time points The one-step predicted mean vector of the upper expanded state vector; At discrete time points The covariance matrix of the expanded state vector is expressed as follows:

[0099] ;

[0100] in, , and All are obtained by time update of the standard Kalman filter (KF):

[0101] ;

[0102] In the formula, At discrete time points The one-step prediction mean of the upper expanded state vector; At discrete time points The estimated value of the state on ; At discrete time points The one-step prediction error covariance matrix of the expanded state vector; At discrete time points The expanded state vector and at discrete time points The cross-covariance matrix between the expanded state vectors; At discrete time points The one-step forecast error covariance matrix on ; At discrete time points The state transfer matrix on ; At discrete time points The process noise covariance matrix on .

[0103] The extended state vector, discrete Bernoulli random variable and delay probability are processed to obtain a posterior probability density function, and a variational Bayes (VB) method is used to obtain a posterior probability density in a free factor form to approximate the joint probability density function.

[0104] The posterior probability density function of the delay probability is expressed as:

[0105] ;

[0106] In the formula, is the posterior probability density function of the delay probability; For a given The expectation of the logarithmic probability density function for all other variables except ; is the set of parameters to be estimated; For a given observation data sequence Under the condition of The posterior probability distribution of ; Represents discrete time points from 1 to All measurements of is a constant. And , Represents a set that contains all the All elements except

[0107] The fixed-point iteration method is used to obtain the approximate form of q(θ). In the i-th iteration, For calculation, we can get the approximate value of q(θ) Based on Bayes' theorem, the joint probability density function It can be written as follows:

[0108] .

[0109] Will , , and Substituting into the above formula, can be restated as:

[0110] .

[0111] Using the actual measured values ​​in the linear state space model and the posterior probability density function, the extended state vector and its error covariance matrix are corrected in the following steps:

[0112] Let θ = , substitute the above formula into The expression is It can be expressed as:

[0113] .

[0114] Using the state expansion method, the posterior probability density function It is expressed as:

[0115] .

[0116] in,

[0117] ;

[0118] ;

[0119] ;

[0120] Posterior probability density function Updated to Gaussian distribution, that is:

[0121] .

[0122] At discrete time points The corrected extended state vector and its error covariance matrix It is expressed as:

[0123] ;

[0124] In the formula, At discrete time points The actual measurement vector after expansion; At discrete time points The expanded measurement matrix above; At discrete time points The upper corrected measurement noise covariance matrix; is the expected value; is the estimated value of the filter state; is the one-step smoothing estimate; is the error covariance matrix corresponding to the filter state estimate; is the cross covariance matrix; is the error covariance matrix corresponding to the one-step smoothing estimate.

[0125] The delay probability and its posterior shape parameters are updated using the modified extended state vector and its error covariance matrix, the steps are as follows:

[0126] set up , then the posterior probability density function is a Bernoulli distribution, expressed as:

[0127] ;

[0128] Among them, in the above formula is a normalization constant, parameter They are given by:

[0129] ;

[0130] expect It is expressed as:

[0131] ;

[0132] set up , then the posterior probability density function is Beta distribution, and the delay probability estimate of the linear system is expressed as:

[0133] ;

[0134] in,

[0135] ;

[0136] Expected Value and The calculations are as follows:

[0137] ;

[0138] In the formula, is the posterior probability density function when the Bernoulli random variable is 1; is the posterior probability density function when the Bernoulli random variable is 0; is an exponential function; is the trace operation of the matrix; At a given discrete time point Observational data on and the current state estimate The expected value of the covariance matrix of the differences between observed and predicted data under the condition of ; At a given discrete time point Observational data on and discrete time points Observational data on The expected value of the covariance matrix of the difference between the observations at two moments under the condition of ; and When the measurement has no delay ( =0), there is a delay in measurement ( =1) when the expected value of the measurement noise covariance matrix; is the number of matrix rows or columns; is the identity matrix; and At discrete time points When the delay probability The shape parameter of the prior Beta distribution of ; is the double gamma function.

[0139] The prediction and update steps are repeated until the preset stopping condition is met, and the delay probability estimate of the linear system is obtained.

[0140] Example 3

[0141] This embodiment introduces a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, the steps of the linear system delay probability estimation method based on the improved Kalman filter algorithm described in Embodiment 1 or 2 are implemented.

[0142] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application may adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.

[0143] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0144] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0145] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0146] The embodiments of the present invention are described above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned specific implementation methods. The above-mentioned specific implementation methods are merely illustrative and not restrictive. Under the enlightenment of the present invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the purpose of the present invention and the claims, which all fall within the protection of the present invention.

Claims

1. A linear system delay probability estimation method based on an improved Kalman filter algorithm, characterized in that: include: Get the state data of the linear system; taking the state data of the linear system as a linear state space model; Adding a discrete Bernoulli random variable to the measurement equation of the linear state space model, and using the probability of the discrete Bernoulli random variable as the delay probability of the linear system; Converting the probability density function of the actual measurement value in the linear state space model into an exponential product form, and constructing a hierarchical Gaussian state space model according to the exponential product form; using the hierarchical Gaussian state space model, expanding the state vector in the linear state space model, and using the discrete Bernoulli random variable and the delay probability as newly added states of the hierarchical Gaussian state space model; The variational Bayes method and the state extension method are used to predict and update the extended state vector, the delay probability and the posterior shape parameters of the delay probability, and the delay probability estimate of the linear system is obtained.

2. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 1 is characterized in that: The linear state space model is expressed as: ； In the formula, is the discrete time index; At discrete time points The state vector on ; At discrete time points The state transfer matrix on ; At discrete time points The state vector on ; At discrete time points The process noise vector on ; At discrete time points The ideal measurement vector on ; At discrete time points The measurement matrix on ; At discrete time points The measurement noise vector on ; At discrete time points The actual measurement vector on ; At discrete time points Discrete Bernoulli random variables on ; At discrete time points The ideal measurement vector on .

3. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 1 is characterized in that: The hierarchical Gaussian state space model is expressed as: ； ； ； In the formula, is a hierarchical Gaussian state space model; It is in the form of exponential product; is the weighted sum form of Gaussian distribution; At discrete time points The actual measurement vector on ; At discrete time points The extended state vector on ; At discrete time points Discrete Bernoulli random variables on ; is a Gaussian distribution; At discrete time points The measurement matrix on ; At discrete time points The state vector on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The measurement matrix on ; At discrete time points The state vector on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The probability of delay on ; is the probability density function of the measurement error when there is a delay in the measurement; is the probability density function of the measurement error when there is no delay in the measurement; is the probability mass function.

4. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 1 is characterized in that: The variational Bayes method and the state extension method are used to predict and update the extended state vector, the delay probability and the posterior shape parameters of the delay probability, and the delay probability estimate of the linear system is obtained, including: Using the forgetting factor to expand the prior shape parameters, calculate the prior probability density function of the delay probability and the one-step prediction mean and covariance matrix of the extended state vector; process the extended state vector, the discrete Bernoulli random variable and the delay probability to obtain the posterior probability density function of the delay probability; The extended state vector and its error covariance matrix are corrected using the actual measured values ​​in the linear state space model and the posterior probability density function, and the delay probability and its posterior shape parameters are updated using the corrected extended state vector and its error covariance matrix to obtain the delay probability estimate of the linear system.

5. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 4 is characterized in that: The prior probability density function of the delay probability is expressed as: ； ； In the formula, is the prior probability density function of the delay probability; is Beta distribution; At discrete time points The probability of delay on ; and At discrete time points Upper delay probability The shape parameter of the prior Beta distribution of ; For the forgetting factor; and At discrete time points Upper delay probability The shape parameter of the posterior Beta distribution of .

6. The linear delay probability estimation method based on the improved Kalman filter algorithm according to claim 4 is characterized in that: The one-step prediction mean and covariance matrix of the extended state vector are expressed as: ； ； In the formula, At discrete time points The one-step predicted mean vector of the upper expanded state vector; At discrete time points The covariance matrix of the expanded state vector; At discrete time points The one-step prediction mean of the upper expanded state vector; At discrete time points The estimated value of the state on ; At discrete time points The one-step prediction error covariance matrix of the expanded state vector; At discrete time points The expanded state vector and at discrete time points The cross-covariance matrix between the expanded state vectors; At discrete time points The one-step forecast error covariance matrix on ; At discrete time points The state transfer matrix on ; At discrete time points The process noise covariance matrix on .

7. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 4 is characterized in that: The posterior probability density function of the delay probability is expressed as: ； In the formula, is the posterior probability density function of the delay probability; For a given The expectation of the logarithmic probability density function for all other variables except ; At discrete time points The extended state vector on ; At discrete time points Discrete Bernoulli random variables on ; At discrete time points The probability of delay on ; is the discrete time index; is the set of parameters to be estimated; For a given observation data sequence Under the condition of The posterior probability distribution of ; Represents discrete time points from 1 to All measurements of is a constant.

8. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 7 is characterized in that: The step of determining the modified extended state vector comprises: set up , using the state expansion method, the posterior probability density function is expressed as: ； in, ； ； ； After updating to Gaussian distribution, the modified extended state vector and its error covariance matrix are expressed as: ； In the formula, is the number of iterations; is a Gaussian distribution; At discrete time points The actual measurement vector after expansion; At discrete time points The expanded measurement matrix above; At discrete time points The upper corrected measurement noise covariance matrix; At discrete time points The one-step predicted mean vector of the upper expanded state vector; At discrete time points The covariance matrix of the expanded state vector; At discrete time points The measurement matrix on ; At discrete time points The measurement matrix on ; At discrete time points The measurement noise covariance matrix on ; At discrete time points The measurement noise covariance matrix on ; is the expected value; At discrete time points The extended state vector after the upper correction; At discrete time points The error covariance matrix after the upper correction; is the estimated value of the filter state; is the one-step smoothing estimate; is the error covariance matrix corresponding to the filter state estimate; is the cross covariance matrix; is the error covariance matrix corresponding to the one-step smoothing estimate.

9. The linear system delay probability estimation method based on the improved Kalman filter algorithm according to claim 8, characterized in that: The step of determining the delay probability estimate of the linear system comprises: set up , then the posterior probability density function is a Bernoulli distribution, expressed as: ； in, ； expect It is expressed as: ； set up , then the posterior probability density function is Beta distribution, and the delay probability estimate of the linear system is expressed as: ； in, ； ； In the formula, is the posterior probability density function when the Bernoulli random variable is 1; is the posterior probability density function when the Bernoulli random variable is 0; is the normalization constant; is an exponential function; is the trace operation of the matrix; At a given discrete time point Observational data on and the current state estimate The expected value of the covariance matrix of the differences between observed and predicted data under the condition of ; At a given discrete time point Observational data on and discrete time points Observational data on The expected value of the covariance matrix of the difference between the observations at two moments under the condition of ; and When the measurement has no delay ( =0), there is a delay in measurement ( =1) when the expected value of the measurement noise covariance matrix; is the number of matrix rows or columns; is the identity matrix; is Beta distribution; and are the shape parameters of the Beta distribution after the i+1th iteration; and At discrete time points When the delay probability The shape parameter of the prior Beta distribution of ; is the double gamma function.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the linear system delay probability estimation method based on the improved Kalman filter algorithm described in any one of claims 1 to 9 are implemented.

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