System Gaussian filter design method for multi-step measurement of random delay and colored measurement of noise

Abstract

The invention discloses a system Gaussian filter design method for multi-step measurement of random delay and colored measurement of noise. A first-order autoregression model is introduced to whiten the colored measurement noise, and a system with the colored measurement noise is converted into a nonlinear system with normal white noise; modeling delay measurement data by means of a Bernoulli random variable, processing multi-step random delay by taking new system measurement noise as a state increment, realizing one-step prediction of delay measurement information, and finally establishing a nonlinear system state model and a measurement model with multi-step measurement random delay and colored measurement noise. And an unscented transformation mode is adopted to approximate non-Gaussian weighted integral, and an unscented Kalman filtering algorithm under the conditions of multi-step measurement random delay and colored measurement noise is provided. According to the method, the accuracy of the filter in processing multi-step measurement random delay and colored measurement noise is improved, the deviation in a filtering result is reduced, and higher precision and better numerical stability are achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and in particular relates to a method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise. Background Art

[0002] Nonlinear state estimation is widely used in target tracking, signal processing, communication, and automatic control. Traditional Gaussian filtering algorithms usually assume that measurement information can arrive on time, and that process noise and measurement noise are Gaussian white noise. However, in actual engineering, due to network congestion, random failure of the transmission mechanism, or limited communication channel bandwidth, the system's measurement values ​​may have random delays of more than two sampling cycles. Traditional solutions usually assume that the delay is fixed, that is, the number of delay steps is consistent at each moment. Although this method is effective in fixed delay scenarios, in actual applications, the number of delay steps often changes randomly, and fixed compensation strategies are difficult to adapt to random delay situations.

[0003] In addition, the classical filtering theory assumes that the measurement noise is Gaussian white noise, that is, the noise sequences are independent, have a mean of zero, and a constant variance. However, in actual systems, the measurement noise often does not meet this assumption and often exhibits time correlation, that is, the characteristics of colored noise. When the measurement noise is colored noise, the classical filter and traditional filtering algorithm will increase the estimation error and the estimation variance because the model assumptions do not match the actual situation, which will significantly reduce the filter performance. Summary of the invention

[0004] The purpose of the present invention is to provide a system Gaussian filter design method for multi-step measurement random delay and colored measurement noise, so as to solve the target tracking problem that the measurement data cannot be updated in time due to the limited bandwidth of the communication channel, and at the same time solve the state estimation problem of the system colored measurement noise.

[0005] The first technical solution adopted by the present invention is: a system Gaussian filter design method for multi-step measurement random delay and colored measurement noise, by introducing a first-order autoregressive model in a nonlinear Gaussian discrete system, using a measurement information amplification method to whiten the colored measurement noise, and then converting the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise, then using independent and identically distributed Bernoulli random variables to model the delayed measurement data, and at the same time using the new system measurement noise as a state increment to achieve a one-step prediction of the delayed measurement information, finally establishing a nonlinear system state model and measurement model with multi-step measurement random delay and colored measurement noise, designing an optimal estimation algorithm in the form of a Gaussian filtering framework, and using an unscented transformation method to approximate a non-Gaussian weighted integral, and providing an unscented Kalman filtering algorithm under the conditions of multi-step measurement random delay and colored measurement noise; the specific operation steps are as follows:

[0006] Step 1, introduce a first-order autoregressive model into the nonlinear Gaussian discrete system, use the measurement information amplification method to whiten the colored measurement noise, and transform the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise;

[0007] Step 2: Use independent and identically distributed Bernoulli random variables and To describe the delay measurement data, determine the multi-step measurement random delay equation, and thus establish a nonlinear system state model and measurement model with multi-step measurement random delay and colored measurement noise;

[0008] Step 3: The new measurement noise ε k Add the system state quantity to the augmented state quantity ξ k Make a one-step prediction to get the one-step prediction of the augmented state at the next moment and the error covariance matrix

[0009] Step 4: According to the measurement model in step 2 and the multi-step measurement random delay equation, the delay y is measured. k+1 Make a prediction to get a one-step prediction of the delay measurement and the error covariance matrix

[0010] Step 5: One-step prediction using the augmented state obtained in steps 3 and 4 One-step prediction of delay measurement The target state is updated through Bayesian filtering to obtain the posterior estimate of the augmented state at the next moment and the error covariance matrix

[0011] Step 6, using unscented transform to approximate the nonlinear Gaussian weighted integral in step 4 and step 5, and then combining the UKF algorithm with the filtering framework under the conditions of multi-step measurement random delay and colored measurement noise proposed above to perform filtering;

[0012] Step 7, repeat steps 2-6 until the entire filtering process is completed.

[0013] The present invention is also characterized in that:

[0014] Step 1 is as follows:

[0015] Step 1.1, introduce the first-order autoregressive model into the nonlinear Gaussian discrete system and give the nonlinear dynamic discrete model of the system:

[0016]

[0017] Among them, x k ∈R n is the target state, z k ∈R m is the ideal measurement value, v k ∈R m is the colored measurement noise of the first-order autoregressive model, expressed as:

[0018]

[0019] in, is the autoregressive coefficient, which describes the influence of the measurement noise value at the previous moment on the current moment, w k With ε k are uncorrelated zero-mean Gaussian white noises, satisfying and Q k and R k Respectively represent w k and ε k The covariance matrix of kl represents the Kronecker function; R n and R m Represent n-dimensional and m-dimensional real vector spaces respectively;

[0020] Step 1.2, use the measurement information amplification method to whiten the colored measurement noise, and transform the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise:

[0021] From formula (1), we get:

[0022] v k =z k -h k (x k ) (3)

[0023] Substituting formula (2) and formula (3) into formula (1), we can obtain:

[0024]

[0025] Using the measurement information amplification method, Then the nonlinear system model with colored measurement noise is:

[0026]

[0027] Obviously, is the new measurement information, is x k+1 and x k Therefore, construct a new state vector:

[0028] Xk+1 =[x k+1 x k ] T (7)

[0029] Then, the nonlinear system model with colored measurement noise, that is, formula (6) is expressed as follows:

[0030]

[0031] Step 2 is specifically implemented as follows:

[0032] Step 2.1: Introduce independent and identically distributed Bernoulli random variables and To describe the s-step delay measurement data, determine the multi-step measurement random delay equation:

[0033]

[0034] Here represents a sequence of Bernoulli random variables with values ​​0 and 1, expressed as follows:

[0035]

[0036] in, is a Bernoulli random variable with probability 0 or 1, is a delay parameter representing the probability of an i-step delay occurring at time k; d = 0, 1, …, s-1, represents the probability of a d-step delay at time k; represents the probability of an s-step delay at time k;

[0037] Step 2.2, according to step 1.2 and step 2.1, establish a nonlinear system state model and a measurement model with multi-step measurement random delay and colored measurement noise:

[0038]

[0039] Step 3 is as follows:

[0040] Step 3.1: Substitute the new measurement noise ε k Add the state quantity to construct a new augmented state vector ξ k :

[0041]

[0042]

[0043] in, is the added measurement noise ε k The state quantity after yes The transposed matrix of yes The transposed matrix of

[0044] Step 3.2, for the augmented state quantity ξ k Make a one-step prediction to get the one-step prediction of the augmented state at the next moment and the error covariance matrix

[0045]

[0046] in, According to the definition, we can get:

[0047]

[0048]

[0049] Among them, Y k-1 is a set of real measurement values;

[0050] according to and The definition of is:

[0051]

[0052] Estimate of the augmented state at the previous moment and the error covariance matrix It is expressed as follows:

[0053]

[0054] Therefore, in formula (18) And in formula (19) can be estimated from the augmented state of the previous moment according to formula (22): and the error covariance matrix Obtained in.

[0055] Step 4 is as follows:

[0056] Step 4.1: Delay measurement y is performed based on the measurement model in step 1.2 and the multi-step measurement random delay equation in step 2.1. k+1 Make a prediction to get a one-step prediction of the delay measurement and the error covariance matrix

[0057]

[0058]

[0059] in, and They represent the one-step prediction and one-step prediction error covariance matrices of the new measurement at time k respectively; and Respectively represent the predicted value and prediction error covariance matrix of the new measurement at time kd; It is the measurement function of the kd moment.

[0060] Step 5 is specifically implemented as follows:

[0061] Step 5.1, one-step prediction using the augmented state obtained in steps 3.2 and 4.1 One-step prediction of delay measurement The target state is updated through Bayesian filtering to obtain the posterior estimate of the augmented state at the next moment and the error covariance matrix

[0062]

[0063] in, represents the augmented state quantity ξ k and delay measurement k The cross-covariance matrix of represents the augmented state quantity ξ k With the new ideal measurement value The cross-covariance matrix of ; Represents the filter gain of the augmented state quantity.

[0064] Step 6 is specifically implemented as follows:

[0065] Step 6.1, use the initial value of the target state to generate a set of determined sigma points, then propagate with nonlinear functions, and finally approximate the mean and error covariance of the state estimate through these sigma points; combine the UKF algorithm with the filtering framework under the conditions of multi-step measurement random delay and colored measurement noise proposed above for filtering, and the specific implementation is as follows:

[0066] Construct sigma point:

[0067]

[0068] Propagate sigma points according to nonlinear system functions:

[0069]

[0070] Where n is the target state x k The dimension of m is the ideal measurement z k The dimension of

[0071] Compute the state forecast mean and its error covariance:

[0072]

[0073] Propagate sigma points according to the nonlinear measurement function:

[0074]

[0075] Approximate calculation formulas (29) to (32) and (37):

[0076]

[0077] Substitute formulas (44) to (48) into the corresponding terms in steps 4.1 and 5.1 to obtain the estimated value of the target state and the error covariance matrix.

[0078] The beneficial effects of the present invention are as follows: the system Gaussian filter design method for multi-step measurement random delay and colored measurement noise proposed in the present invention designs an optimal estimation algorithm in the form of a Gaussian filter framework, and adopts an unscented transformation method to approximate the non-Gaussian weighted integral, and provides an unscented Kalman filter algorithm under the conditions of multi-step measurement random delay and colored measurement noise. This method can solve the target tracking problem where the measurement data cannot be updated in time due to the limited bandwidth of the communication channel, and can also solve the state estimation problem of the system measurement noise that is non-Gaussian white noise. The estimation effect is optimized, making the effect more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 It is a global flow chart of the Gaussian filter design method of the present invention for multi-step measurement of nonlinear systems under random delay and colored measurement noise conditions;

[0080] Figure 2 is a velocity root mean square error diagram of the target state estimation using the UKF algorithm with one-step measurement random delay and colored measurement noise and the method of the present invention in Embodiment 6 of the present invention respectively;

[0081] Figure 3 It is a position root mean square error diagram of the target state estimation using the UKF algorithm with one-step measurement random delay and colored measurement noise and the method of the present invention in Example 6 of the present invention. DETAILED DESCRIPTION

[0082] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments so that the advantages and features of the present invention can be more easily understood by those skilled in the art.

[0083] Example 1

[0084] The invention discloses a method for designing a system Gaussian filter with multi-step measurement random delay and colored measurement noise. In a nonlinear Gaussian discrete system, a first-order autoregressive model is introduced, and the colored measurement noise is whitened by a measurement information amplification method, thereby converting the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise. Then, independent and identically distributed Bernoulli random variables are used to model the delayed measurement data. At the same time, the new system measurement noise is used as a state increment to realize a one-step prediction of the delayed measurement information. Finally, a nonlinear system state model and a measurement model with multi-step measurement random delay and colored measurement noise are established, an optimal estimation algorithm in the form of a Gaussian filtering framework is designed, and an unscented transformation method is used to approximate a non-Gaussian weighted integral. An unscented Kalman filtering algorithm under the conditions of multi-step measurement random delay and colored measurement noise is given.

[0085] Example 2

[0086] The Gaussian filter design method for multi-step measurement of random delay and colored measurement noise of the present invention has the following specific operation steps:

[0087] Step 1, introduce a first-order autoregressive model into the nonlinear Gaussian discrete system, use the measurement information amplification method to whiten the colored measurement noise, and transform the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise;

[0088] Step 2: Use independent and identically distributed Bernoulli random variables and To describe the delay measurement data, determine the multi-step measurement random delay equation, and thus establish a nonlinear system state model and measurement model with multi-step measurement random delay and colored measurement noise;

[0089] Step 3: The new measurement noise ε k Add the system state quantity to the augmented state quantity ξ k Make a one-step prediction to get the one-step prediction of the augmented state at the next moment and the error covariance matrix

[0090] Step 4: According to the measurement model in step 2 and the multi-step measurement random delay equation, the delay y is measured. k+1 Make a prediction to get a one-step prediction of the delay measurement and the error covariance matrix

[0091] Step 5: One-step prediction using the augmented state obtained in steps 3 and 4 One-step prediction of delay measurement The target state is updated through Bayesian filtering to obtain the posterior estimate of the augmented state at the next moment and the error covariance matrix

[0092] Step 6, using unscented transform to approximate the nonlinear Gaussian weighted integral in step 4 and step 5, and then combining the UKF algorithm with the filtering framework under the conditions of multi-step measurement random delay and colored measurement noise proposed above to perform filtering;

[0093] Step 7, repeat steps 2-6 until the entire filtering process is completed.

[0094] Example 3

[0095] Based on Example 2, step 1 is as follows:

[0096] Step 1.1, introduce the first-order autoregressive model into the nonlinear Gaussian discrete system and give the nonlinear dynamic discrete model of the system:

[0097]

[0098] Among them, x k ∈R n is the target state, z k ∈R m is the ideal measurement value, v k ∈R m is the colored measurement noise of the first-order autoregressive model, expressed as:

[0099]

[0100] in, is the autoregressive coefficient, which describes the influence of the measurement noise value at the previous moment on the current moment, w k With ε k are uncorrelated zero-mean Gaussian white noises, satisfying and Q k and R k Respectively represent w k and ε k The covariance matrix of kl represents the Kronecker function; R n and R m Represent n-dimensional and m-dimensional real vector spaces respectively;

[0101] Step 1.2, use the measurement information amplification method to whiten the colored measurement noise, and transform the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise:

[0102] From formula (1), we get:

[0103] v k =z k -h k (x k ) (3)

[0104] Substituting formula (2) and formula (3) into formula (1), we can obtain:

[0105]

[0106] Using the measurement information amplification method, Then the nonlinear system model with colored measurement noise is:

[0107]

[0108] Obviously, is the new measurement information, is x k+1 and x k Therefore, construct a new state vector:

[0109] X k+1 =[x k+1 x k ] T (7)

[0110] Then, the nonlinear system model with colored measurement noise, that is, formula (6) is expressed as follows:

[0111]

[0112] Step 2 is specifically implemented as follows:

[0113] Step 2.1: Introduce independent and identically distributed Bernoulli random variables and To describe the s-step delay measurement data, determine the multi-step measurement random delay equation:

[0114]

[0115] Here represents a sequence of Bernoulli random variables with values ​​0 and 1, expressed as follows:

[0116]

[0117] in, is a Bernoulli random variable with probability 0 or 1, is a delay parameter representing the probability of an i-step delay occurring at time k; d = 0, 1, …, s-1, represents the probability of a d-step delay at time k; represents the probability of an s-step delay at time k;

[0118] Step 2.2, according to step 1.2 and step 2.1, establish a nonlinear system state model and a measurement model with multi-step measurement random delay and colored measurement noise:

[0119]

[0120] Example 4

[0121] Based on Example 3, step 3 is as follows:

[0122] Step 3.1: Substitute the new measurement noise ε k Add the state quantity to construct a new augmented state vector ξ k :

[0123]

[0124] in, is the added measurement noise ε k The state quantity after yes The transposed matrix of yes The transposed matrix of

[0125] Step 3.2, for the augmented state quantity ξ k Make a one-step prediction to get the one-step prediction of the augmented state at the next moment and the error covariance matrix

[0126]

[0127] in, According to the definition, we can get:

[0128]

[0129] Among them, Y k-1 is a set of real measurement values;

[0130] according to and The definition of is:

[0131]

[0132]

[0133] Estimate of the augmented state at the previous moment and the error covariance matrix It is expressed as follows:

[0134]

[0135] And in formula (19) can be estimated from the augmented state of the previous moment according to formula (22): and the error covariance matrix Obtained in.

[0136] Step 4 is as follows:

[0137] Step 4.1: Delay measurement y is performed based on the measurement model in step 1.2 and the multi-step measurement random delay equation in step 2.1. k+1 Make a prediction to get a one-step prediction of the delay measurement and the error covariance matrix

[0138]

[0139] in, and They represent the one-step prediction and one-step prediction error covariance matrices of the new measurement at time k respectively; and Respectively represent the predicted value and prediction error covariance matrix of the new measurement at time kd; It is the measurement function of the kd moment.

[0140] Step 5 is specifically implemented as follows:

[0141] Step 5.1, one-step prediction using the augmented state obtained in steps 3.2 and 4.1 One-step prediction of delay measurement The target state is updated through Bayesian filtering to obtain the posterior estimate of the augmented state at the next moment and the error covariance matrix

[0142]

[0143] in, represents the augmented state quantity ξ k and delay measurement k The cross-covariance matrix of represents the augmented state quantity ξ k With the new ideal measurement value The cross-covariance matrix of ; Represents the filter gain of the augmented state quantity.

[0144] Example 5

[0145] Based on Example 4, step 6 is specifically implemented in the following manner:

[0146] Step 6.1, use the initial value of the target state to generate a set of determined sigma points, then propagate with nonlinear functions, and finally approximate the mean and error covariance of the state estimate through these sigma points; combine the UKF algorithm with the filtering framework under the conditions of multi-step measurement random delay and colored measurement noise proposed above for filtering, and the specific implementation is as follows:

[0147] Construct sigma point:

[0148]

[0149] Propagate sigma points according to nonlinear system functions:

[0150]

[0151] Where n is the target state x k The dimension of m is the ideal measurement z k The dimension of

[0152] Compute the state forecast mean and its error covariance:

[0153]

[0154] Propagate sigma points according to the nonlinear measurement function:

[0155]

[0156] Approximate calculation formulas (29) to (32) and (37):

[0157]

[0158] Substitute formulas (44) to (48) into the corresponding terms in steps 4.1 and 5.1 to obtain the estimated value of the target state and the error covariance matrix.

[0159] Example 6

[0160] The present invention uses traceless transformation to numerically calculate the Gaussian weighted integral in the proposed framework. In order to evaluate the performance of the method of the present invention, we select a nonlinear system with two-step random delay measurement as an example, and use UKF with one-step measurement random delay and synchronous correlated noise and UKF with two-step measurement random delay and synchronous correlated noise to process the state estimation of the nonlinear system with two-step random delay measurement. The system equation and measurement equation of the nonlinear system are as follows:

[0161]

[0162] Among them, the state vector x k and k Represents the position in the x and y directions respectively, and denote the speed in the x and y directions respectively, Ω denotes the constant and unknown turning rate, T is the sampling time interval, and the process noise w k ~N(0,Q k ) is Gaussian white noise; z k is the ideal measurement at time k, and the measurement noise v k According to the formula (2) in Example 3, the white noise ε k ~N(0,R k ) is Gaussian white noise; Q k With R k Satisfy the following formula:

[0163]

[0164] R k =diag{100m,10m·rad}

[0165] Set the initial values ​​of the simulation parameters:

[0166] T=1s,η 1 =0.1m 2 ·s -3

[0167] x 0 =[1005m 10m·s -1 4020m 50m·s -1 ] T

[0168] P 0|0 =diag{100m 2 ,10m 2 ·s -2 ,100m 2 ,10m 2 ·s -2}

[0169] Autoregressive coefficient The measured probability of no delay is The probability of one-step measurement delay is The probability of two-step measurement delay is Delay parameter p 1 =p 2 =0.5, delay step number s=2;

[0170] In the simulation, the root mean square error (RMSE) of the state estimation result is used as the performance indicator of the evaluation algorithm, which is defined as follows:

[0171]

[0172] Where, N represents the number of Monte Carlo simulations, and N is set to 250;

[0173] According to Figure 1 The process execution steps are as follows:

[0174] Execute step 1, initialize relevant parameters and whiten the colored measurement noise using a measurement information amplification method;

[0175] Execute step 2 to establish a target state model and a measurement model under the conditions of multi-step measurement random delay and colored measurement noise according to the initial value in step 1;

[0176] Execute step 3 to predict the target state and the covariance of the target state at the next moment to obtain the predicted value and

[0177] Execute step 4 to calculate the actual measurement y at the next moment according to the measurement model and multi-step measurement random delay equation in step 2. k+1 Perform a one-step prediction to obtain a one-step prediction of the delay measurement and the one-step forecast error covariance matrix

[0178] Execute step 5, use the predicted value of the target state obtained in steps 3 and 4 and the delayed measurement prediction to estimate the target state through Bayesian filtering, and obtain the posterior estimated value of the target state at the next moment and the error covariance matrix

[0179] Execute step 6, using an unscented transform to approximate the nonlinear Gaussian weighted integrals in steps 4 and 5;

[0180] Go to step 7 and repeat steps 2 to 6 250 times until the whole process is completed;

[0181] Step 8, the final result is as follows Figure 2-3 shown.

[0182] from Figure 2 and Figure 3It can be clearly seen that when the measurement noise is colored and the system measurement is randomly delayed by two sampling times, the method of the present invention has higher estimation accuracy than the UKF with one-step measurement random delay, and its position root mean square error and velocity root mean square error are both smaller. Therefore, compared with the existing method, the method of the present invention is more suitable for realizing state estimation of nonlinear systems with multi-step measurement random delays and colored measurement noise.

Claims

1. A method for designing a Gaussian filter for a system with multi-step measurement random delay and colored measurement noise, characterized in that: In the nonlinear Gaussian discrete system, a first-order autoregressive model is introduced and the colored measurement noise is whitened by the measurement information amplification method. Then, the nonlinear dynamic discrete system with colored measurement noise is transformed into a nonlinear system with normal white noise. Then, the delayed measurement data is modeled with the help of independent and identically distributed Bernoulli random variables. At the same time, the new system measurement noise is used as the state increment to realize one-step prediction of the delayed measurement information. Finally, the state model and measurement model of the nonlinear system with multi-step measurement random delay and colored measurement noise are established, an optimal estimation algorithm in the form of a Gaussian filtering framework is designed, and the unscented transformation method is used to approximate the non-Gaussian weighted integral. The unscented Kalman filtering algorithm under the conditions of multi-step measurement random delay and colored measurement noise is given.

2. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 1, characterized in that: Step 1, introduce a first-order autoregressive model into the nonlinear Gaussian discrete system, use the measurement information amplification method to whiten the colored measurement noise, and transform the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise; Step 2: Use independent and identically distributed Bernoulli random variables and To describe the delay measurement data, determine the multi-step measurement random delay equation, and thus establish a nonlinear system state model and measurement model with multi-step measurement random delay and colored measurement noise; Step 3: The new measurement noise ε k Add the system state quantity to the augmented state quantity ξ k Make a one-step prediction to get the one-step prediction of the augmented state at the next moment and the error covariance matrix Step 4: According to the measurement model in step 2 and the multi-step measurement random delay equation, the delay y is measured. k+1 Make a prediction to get a one-step prediction of the delay measurement and the error covariance matrix Step 5: One-step prediction using the augmented state obtained in steps 3 and 4 One-step prediction of delay measurement The target state is updated through Bayesian filtering to obtain the posterior estimate of the augmented state at the next moment and the error covariance matrix Step 6, using unscented transform to approximate the nonlinear Gaussian weighted integral in step 4 and step 5, and then combining the UKF algorithm with the filtering framework under the conditions of multi-step measurement random delay and colored measurement noise proposed above to perform filtering; Step 7, repeat steps 2-6 until the entire filtering process is completed.

3. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 2, characterized in that: Step 1 is as follows: Step 1.1, introduce the first-order autoregressive model into the nonlinear Gaussian discrete system and give the nonlinear dynamic discrete model of the system: Among them, x k ∈R n is the target state, z k ∈R m is the ideal measurement value, v k ∈R m is the colored measurement noise of the first-order autoregressive model, expressed as: in, is the autoregressive coefficient, which describes the influence of the measurement noise value at the previous moment on the current moment, w k With ε k are uncorrelated zero-mean Gaussian white noises, satisfying and Q k and R k Respectively represent w k and ε k The covariance matrix of kl represents the Kronecker function; R n and R m Represent n-dimensional and m-dimensional real vector spaces respectively; Step 1.2, use the measurement information amplification method to whiten the colored measurement noise, and transform the nonlinear dynamic discrete system with colored measurement noise into a nonlinear system with normal white noise: From formula (1), we get: v k =z k -h k (x k ) (3) Substituting formula (2) and formula (3) into formula (1), we can obtain: Using the measurement information amplification method, Then the nonlinear system model with colored measurement noise is: Obviously, is the new measurement information, is x k+1 and x k Therefore, construct a new state vector: X k+1 =[x k+1 x k ] T (7) Then, the nonlinear system model with colored measurement noise, that is, formula (6) is expressed as follows:

4. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 3, characterized in that: Step 2 is specifically implemented as follows: Step 2.1: Introduce independent and identically distributed Bernoulli random variables and To describe the s-step delay measurement data, determine the multi-step measurement random delay equation: Here represents a sequence of Bernoulli random variables with values ​​0 and 1, expressed as follows: in, is a Bernoulli random variable with probability 0 or 1, is a delay parameter representing the probability of an i-step delay occurring at time k; d = 0, 1, …, s-1, represents the probability of a d-step delay at time k; represents the probability of an s-step delay at time k; Step 2.2, according to step 1.2 and step 2.1, establish a nonlinear system state model and a measurement model with multi-step measurement random delay and colored measurement noise:

5. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 4, characterized in that: Step 3 is as follows: Step 3.1: Substitute the new measurement noise ε k Add the state quantity to construct a new augmented state vector ξ k : in, is the added measurement noise ε k The state quantity after yes The transposed matrix of yes The transposed matrix of Step 3.2, for the augmented state quantity ξ k Make a one-step prediction to get the one-step prediction of the augmented state at the next moment and the error covariance matrix in, According to the definition, we can get: Among them, Y k-1 is a set of real measurement values; according to and The definition of is: Estimate of the augmented state at the previous moment and the error covariance matrix It is expressed as follows: Therefore, in formula (18) And in formula (19) can be estimated from the augmented state of the previous moment according to formula (22): and the error covariance matrix Obtained in.

6. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 5, characterized in that: Step 4 is as follows: Step 4.1: Delay measurement y is performed based on the measurement model in step 1.2 and the multi-step measurement random delay equation in step 2.

1. k+1 Make a prediction to get a one-step prediction of the delay measurement and the error covariance matrix in, and They represent the one-step prediction and one-step prediction error covariance matrices of the new measurement at time k respectively; and Respectively represent the predicted value and prediction error covariance matrix of the new measurement at time kd; It is the measurement function of kd moment.

7. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 6, characterized in that: Step 5 is specifically implemented as follows: Step 5.1, one-step prediction using the augmented state obtained in steps 3.2 and 4.1 One-step prediction of delay measurement The target state is updated through Bayesian filtering to obtain the posterior estimate of the augmented state at the next moment and the error covariance matrix in, represents the augmented state quantity ξ k and delay measurement k The cross-covariance matrix of represents the augmented state quantity ξ k With the new ideal measurement value The cross-covariance matrix of ; Represents the filter gain of the augmented state quantity.

8. The method for designing a system Gaussian filter for multi-step measurement of random delay and colored measurement noise according to claim 6, characterized in that: Step 6 is specifically implemented as follows: Step 6.1, use the initial value of the target state to generate a set of determined sigma points, then propagate with nonlinear functions, and finally approximate the mean and error covariance of the state estimate through these sigma points; combine the UKF algorithm with the filtering framework under the conditions of multi-step measurement random delay and colored measurement noise proposed above for filtering, and the specific implementation is as follows: (1) Construct sigma point: (2) Propagation of sigma points according to nonlinear system functions: Where n is the target state x k The dimension of m is the ideal measurement z k The dimension of (3) Calculate the state prediction mean and its error covariance: (4) Propagation of sigma points according to nonlinear measurement function: (5) Approximate calculation formulas (29) to (32) and (37): Substitute formulas (44) to (48) into the corresponding terms in steps 4.1 and 5.1 to obtain the estimated value of the target state and the error covariance matrix.

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